Suppose that A is a subset of\mathbb{N}and(1) 0,1 ∈ A(2) if n ∈ A, then 4n ∈ A.Give a careful proof that {4n : n ∈\mathbb{N}} is a subset of A. (Apply induction on n.)

Answer:

y <= x+2

Step-by-step explanation:

Finding the curve equation,

the slope is 1 and the y-intercept is 2. Hence,

y = x + 2

Since the thing is under the graph,

y < x + 2

Since it is a solid line,

y <= x + 2

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One possible absolute value inequality with the solution set 5 is:

| x - 5 | ≤ 0

An absolute value inequality is a type of inequality that involves the absolute value of a variable. The absolute value of a number is its distance from zero, and it is always a non-negative value.

The general form of an absolute value inequality is:

| f(x) | < a

where f(x) is an algebraic expression involving x, and a is a positive number.

An absolute value inequality with the solution set of 5 can be written in the form:

| x - b | ≤ c

where b is the value around which x can vary and c is the maximum distance from b to the boundary of the solution set.

To obtain a solution set of 5, we need to choose b as the midpoint between the two endpoints of the solution set, which is (5 + 5)/2 = 5.

The distance from b to either endpoint of the solution set is 5 - 5 = 0. Therefore, we can choose c to be any value greater than or equal to 0.

One possible absolute value inequality with the solution set 5 is:

| x - 5 | ≤ 0

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The limit lim(x→1) (x^a - 1)(x^b - 1) is equal to ab.

To evaluate the limit lim(x→1) (x^a - 1)(x^b - 1), we'll follow these steps:

1. Recognize the given expression: (x^a - 1)(x^b - 1)
2. Apply the limit: lim(x→1) (x^a - 1)(x^b - 1)
3. Factor using the difference of squares: (x - 1)(x^(a-1) + x^(a-2) + ... + 1)(x - 1)(x^(b-1) + x^(b-2) + ... + 1)
4. Cancel out the common factor of (x - 1) in both terms: lim(x→1) (x^(a-1) + x^(a-2) + ... + 1)(x^(b-1) + x^(b-2) + ... + 1)
5. Substitute x = 1 in the remaining expression: (1^(a-1) + 1^(a-2) + ... + 1)(1^(b-1) + 1^(b-2) + ... + 1)
6. Simplify: (1 + 1 + ... + 1)(1 + 1 + ... + 1)
7. Count the number of terms in each parenthesis and multiply them.

Since there are "a" terms in the first parentheses and "b" terms in the second parentheses, the final answer is ab.
So, the limit lim(x→1) (x^a - 1)(x^b - 1) is equal to ab.

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The sequence an=(9n)/√(n^2+1) converges to 9/sqrt(1+1)=9/sqrt(2)=9√2/2, so the answer is (a) converges to (9√2)/2.

To see why, we can use the limit comparison test, comparing to a similar sequence bn = 9n/sqrt(n^2), which simplifies to bn = 9/sqrt(n). Since the limit as n approaches infinity of bn is 0, we can use this to find the limit of an by taking the limit of the ratio an/bn:

lim(n->inf) an/bn = lim(n->inf) [(9n)/√(n^2+1)] / [9/sqrt(n)]
= lim(n->inf) sqrt(n) * (n/sqrt(n^2+1))
= lim(n->inf) (n/sqrt(n^2+1)) (since sqrt(n) approaches infinity as n approaches infinity)
= lim(n->inf) (1/sqrt(1+(1/n^2))) = 1/sqrt(1+0) = 1/sqrt(1) = 1.

Since this limit is finite and nonzero, we can conclude that the sequence converges, and its limit is 9/sqrt(2).

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5x - y - 13 = 0  is the equation of the plane

To find the equation of the plane, we need two pieces of information: a point on the plane, and the normal vector of the plane.

We are given a point on the plane: P0(2, -3, -5).

To find the normal vector of the plane, we need to use the fact that the plane is perpendicular to the given line. The direction vector of the line is <1, 5, 0>, since the line has parametric equations x=2+t, y=-3+5t, z=-3, and the vector <1, 5, 0> is the coefficient vector of the parameter t in these equations.

Any vector that is perpendicular to <1, 5, 0> will be a normal vector of the plane. One such vector is <5, -1, 0>, which we can verify by taking the dot product of this vector with <1, 5, 0>:

<5, -1, 0> · <1, 5, 0> = 5(1) + (-1)(5) + 0(0) = 0

Thus, the normal vector of the plane is <5, -1, 0>.

Now we can use the point-normal form of the equation of a plane:

ax + by + cz = d

where <a, b, c> is the normal vector of the plane, and (x, y, z) is any point on the plane. Substituting in the values we have:

5(x - 2) - 1(y + 3) + 0(z + 5) = 0

Simplifying:

So the equation of the plane is:

5x - y - 13 = 0

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The statement that is true about the quadrilateral is

It is a rectangle because the opposite sides in a quadrilateral are congruent.

Option C is the correct answer.

We have,

From the given information,

AB = CD = √73

BC = AD = √5

Slope:

BA = CD = 1/2

BC = AD = 8/3

Now,

This means,

AB = CD = congruent

BC = AD = congruent

Now,

The opposite sides in a quadrilateral are congruent.

This means,

The quadrilateral is a rectangle.

Thus,

The statement that is true about the quadrilateral is

It is a rectangle because the opposite sides in a quadrilateral are congruent.

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To sketch the vector field F(r) = -r / ||r||^3 in the xy-plane, we can first observe that this is a radial vector field that points towards the origin. As ||r||^3 is the cube of the distance from the origin, the denominator increases much faster than the numerator, causing the lengths of the vectors to decrease as we move away from the origin. Therefore, the first statement "The lengths of the vectors decrease as you move away from the origin. All the vectors point towards the origin" is true.

As for the second statement, "The length of each vector is 1. All the vectors point in the same direction. All the vectors point away from the origin", it is not true for this vector field. The length of each vector depends on the distance from the origin and is not constant. Also, the vectors point towards the origin and not away from it. Therefore, this statement is false.

In summary, the correct answer is: The lengths of the vectors decrease as you move away from the origin. All the vectors point towards the origin.

To sketch the vector field F(r) = -r / ||r||^3 in the xy-plane and determine which statements apply, follow these steps:

1. Recognize that F(r) is a radial vector field with its direction determined by the term -r, which points towards the origin, and its magnitude determined by 1/||r||^3.

2. Notice that as you move away from the origin (increasing the value of ||r||), the magnitude of the vector field decreases because the denominator ||r||^3 increases, making the overall value of the vector field smaller.

3. Observe that all vectors point towards the origin because of the negative sign in the term -r.

4. Since the magnitude of the vector field is determined by 1/||r||^3 and not a constant value, the length of each vector is not 1.

5. As the vector field is radial and determined by the term -r, the vectors do not point in the same direction and do not point away from the origin.

From this analysis, we can conclude that the following statements apply:

- The lengths of the vectors decrease as you move away from the origin.
- All the vectors point towards the origin.

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The number of pages with photos rounded to the nearest whole number is 204 pages.

First, we need to find out how many pages of the album are empty. Since 59% of the album is empty, that means 41% of the album is filled with photos.

To find out how many photos are in the album, we multiply the number of pages by the number of photos per page:

500 pages x 5 photos per page = 2500 photos

To find out how many pages are filled with photos, we need to take 41% of the total number of pages:

500 pages x 0.41 = 205 pages

However, since we're looking for the number of pages with photos rounded to the nearest whole number, we round down to 204 pages. Therefore, each of the 18 students would receive 204/18 = 11.33 pages of photos (rounded to the nearest hundredth).

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Complete Question:

Roy has an album that holds. 500 Each page of the album holds 5 photos. If 59​% of the album is​ empty, how many pages are filled with photos​?

The answer is (a) [tex]$-\frac{9}{2}$[/tex].

To find the area of the bounded region determined by the curves [tex]$x^2=8y[/tex]and x + 4y - 4 = 0, we first need to find the intersection points of the two curves.

From the equation [tex]$x^2=8y$[/tex], we get [tex]$y=\frac{x^2}{8}$[/tex] Substituting this in the equation x + 4y - 4 = 0, we get [tex]$x+4\left(\frac{x^2}{8}\right)-4=0$[/tex], which simplifies to [tex]$x^2+8x-32=0$[/tex]. Solving for x, we get [tex]$x=-4\pm 4\sqrt{3}$[/tex].

Since the parabola [tex]$x^2=8y$[/tex] opens upwards, the area of the bounded region can be calculated as follows:

[tex]Area }=\int_{-4-4 \sqrt{3}}^{4 \sqrt{2}} \int_{\frac{x^2}{8}}^{(4-x) / 4} d y d x[/tex]

Integrating with respect to y first, we get:

[tex]\text { Area }=\int_{-4-4 \sqrt{3}}^{4 \sqrt{2}}\left(\frac{4-x}{4}-\frac{x^2}{8}\right) d x[/tex]

Simplifying and evaluating the integral, we get:

[tex]\text { Area }=\frac{9}{2}+\frac{16 \sqrt{3}}{3}-2 \sqrt{2}[/tex]

Therefore, the answer is (a)[tex]$-\frac{9}{2}$[/tex].

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To calculate the probability that none of the 60 randomly selected subjects left the treatment center against medical advice, we will use some steps.

Those steps are:
1. Calculate the probability of a single subject not leaving against medical advice.
2. Calculate the probability of all 60 subjects not leaving against medical advice.

Step 1:
There are a total of 55,673 patients, out of which 55,423 did not leave against medical advice. So, the probability of a single subject not leaving against medical advice is:
P(not leaving) = (number of patients not leaving) / (total number of patients)
P(not leaving) = 55,423 / 55,673 ≈ 0.9955

Step 2:
Since the subjects are randomly selected without replacement, we need to adjust the probability for each subsequent selection. However, as the sample size (60) is much smaller than the total number of patients (55,673), the difference in probabilities will be negligible. Therefore, we can assume that the probability for each subject remains approximately the same.

To calculate the probability that none of the 60 subjects left the treatment center against medical advice, we will multiply the probability of each subject not leaving against medical advice:

P(all 60 not leaving) = (P(not leaving))^60
P(all 60 not leaving) = (0.9955)^60 ≈ 0.7409

So, the probability that none of the 60 randomly selected subjects left the treatment center against medical advice is approximately 0.7409 or 74.09%.

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Answer:

C is correct in case of division

Step-by-step explanation:

A is correct in case of addition

B is correct in case of multiplication

Answer:C

Step-by-step explanation: take the exponent value in the numerator  and subtract the exponent value of the denominator

Jamal would need approximately 7.9 gallons of gas to travel 296.25 miles.

We can use the concept of direct variation to solve this problem. Direct variation means that two quantities are related by a constant ratio. In this case, the number of miles driven is directly proportional to the number of gallons used.

To find the constant of proportionality, we can use the given data. From the table, we can see that when Jamal used 14 gallons, he drove 525 miles. So we can write:

14/525 = k

where k is the constant of proportionality.

Solving for k, we get:

k = 14/525

Now we can use this value of k to find how many gallons Jamal would need to travel 296.25 miles. Let x be the number of gallons he would need. Then we can write:

x/296.25 = k

Substituting the value of k, we get:

x/296.25 = 14/525

Solving for x, we get:

x = (296.25 × 14) / 525

x ≈ 7.9

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Answer:

The slope is -3

Step-by-step explanation:

Select two points on the line.  I have selected the points (1,5) and (2,2).  The slope is the change in y over the change in x.  The y values are 2 and 5,  The x values are 2 and 1.  You find the change by subtracting.

[tex]\frac{2-5}{2-1}[/tex] = [tex]\frac{-3}{1}[/tex] = -3

Another way to look at this is seeing the slope as the rise over the run.  If you start at (1,5) and only move right or left and up and down to get to (2,2), you would have to move straight down 3 spaces and then 1 space right.  Down is negative and right is positive, so the slope would be [tex]\frac{-3}{1}[/tex] which equals -3.

Helping in the name of Jesus.

The volume of the given solid is 12 cubic units.

To evaluate the given double integral, we can first identify it as the volume of a solid. In this case, the integrand is (16-8y), which represents the height of the solid at any given point (x,y) in the region R=[0,1]x[0,1].

Thus, to find the volume of this solid, we need to integrate this height function over the entire region R.

∫ ∫ R (16 − 8y)dA = ∫₀¹ ∫₀¹ (16-8y) dx dy

Evaluating this double integral using iterated integration, we get:

∫₀¹ ∫₀¹ (16-8y) dx dy = ∫₀¹ [16x - 8yx] from x=0 to x=1 dy

= ∫₀¹ (16-8y) dy

= [16y - 4y²] from y=0 to y=1

= (16-4) - (0-0)

= 12

Therefore, the volume of the given solid is 12 cubic units.

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The volume of the solid lying under the plane 8-2x-y is 32 cubic units.

To find the volume of the solid lying under the plane 8-2x-y, you need to use a triple integral.

To find the volume, you need to perform a triple integral over the region, integrating the function 8-2x-y with respect to x, y, and z. First, find the limits of integration for x, y, and z by determining the intersections of the plane with the coordinate axes. The intersections are (4,0,0), (0,8,0), and (0,0,8). Next, set up the triple integral as follows:

∭(8-2x-y)dzdydx, with x ranging from 0 to 4, y ranging from 0 to 8-2x, and z ranging from 0 to 8-2x-y.

Evaluate the integral with respect to z first, then y, and finally x. After evaluating, you will find that the volume of the solid is 32 cubic units.

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David gained approximately $2,155.06 in interest after 4 years.

By utilizing the compound interest formula A = P(1 + r/n)^(nt), one can determine the future value of an investment or loan, inclusive of its added interest.

Variables to consider include the initial deposit (P), annual interest rate (r as a decimal), frequency at which it is compounded per year (n) and time (t).

This specific scenario assimilates a principal amount of $10,000 with an annual interest rate of 5% compounded yearly for four years, resulting in an accrued balance of roughly $12,155.06.

Therefore, David gained approximately $2,155.06 in interest after 4 years.

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If David deposited $10,000 into a bank account with a 5% interest rate compounded annually, how much interest did he gain after 4 years?

Now, we can evaluate the improper integral of f(x) from 1 to infinity:
integral^infinity_1 1/(2x + 8) dx. Since, the improper integral diverges, the series also diverges by the Integral Test. Therefore, the answer is b. diverges.

Your answer: b. diverges

To apply the Integral Test, we first need to confirm that the function f(x) = 1/(2x + 8) is continuous, positive, and decreasing on the interval [1, ∞). Since the function meets these conditions, we can apply the Integral Test.

Now, let's evaluate the integral:
∫[1,∞] (1/(2x + 8)) dx

To solve this, we can use the substitution method:

let u = 2x + 8, so du = 2 dx. Now, when x = 1, u = 10, and when x → ∞, u → ∞.
Now, the integral becomes:
(1/2) ∫[10,∞] (1/u) du

This is an improper integral, and its form is a p-series where p = 1. We know that a p-series converges if p > 1 and diverges if p ≤ 1. In this case, p = 1, so the integral diverges.
Since the integral diverges, by the Integral Test, the given series also diverges.

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In a case whereby Mai poured 2.4 L into a partilly filled water now there is 10.4 the best figure that represent this is the second fiqure.

Based on the given information it can be seen that the total volume of the figure is 10.4 which implies that it will take the total volume of of water of 10.4

Considering the second fiqure , it can be deduced that the total volume is 10.4, where one part of the fiqure is X and other bis 2.4, which impies that 10.4 = X + 2.4 which is the expression for the fiqure.

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answer: 9.1

The points are plotted at (-4,6) and (5,5)

i have attatched a photo with the values inputted into the distance formula!

Answer:

The highlighted part is the answer

Step-by-step explanation:

Mark needs to add heat to the solution in order to elevate the temperature from -4.3°C to a positive value

The amount of heat needed to increase a substance's temperature by one degree Celsius per gramme is known as its specific heat capacity. It is a characteristic of a substance that is intense and independent of the size or shape of the quantity under consideration. A substance's specific heat capacity is typically indicated by the letters "c" or "s"².

Mark needs to add heat to the solution in order to elevate the temperature from -4.3°C to a positive value.

. The bulk of the solution and its specific heat capacity affect the amount of heat needed.

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The best graph to display the data is C. a bar graph titled favorite topping with the x axis labeled topping and the y axis labeled number of customers, with the first bar labeled chocolate chips going to a value of 17, the second bar labeled hot fudge going to a value of 12, the third bar labeled nuts going to a value of 27, and the fourth bar labeled sprinkles going to a value of 44.

Categorical data is best represented through a bar graph wherein every distinctive category is illustrated by a rectangular bar in correspondence to the frequency or item count. This display method uses the height or length of each rectangle as its basis.

The customer's preferences on their choice of ice cream toppings were subjected to a categorical survey; hence, it deemed suitable for visual illustration via a bar chart. There are four recognizable component ingredients identified in this survey namely: Sprinkles, Nuts, Hot Fudge, and Chocolate Chips. Each component underlies particular statistical value and described discretely hence a bar graph proves fitting to portray this information.

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As R² with the given operations satisfies all the vector space axioms, it is indeed a vector space.

To determine whether the set R² with the given operations is a vector space, we need to verify if it satisfies all the vector space axioms.

1. Closure under addition: (x1, y1) + (x2, y2) = (x1 + x2, y1 + y2), which is of the same form as the original elements in R². Thus, addition is closed.

2. Commutativity of addition: (x1, y1) + (x2, y2) = (x1 + x2, y1 + y2) = (x2 + x1, y2 + y1) = (x2, y2) + (x1, y1). Thus, addition is commutative.

3. Associativity of addition: ((x1, y1) + (x2, y2)) + (x3, y3) = (x1 + x2, y1 + y2) + (x3, y3) = (x1 + x2 + x3, y1 + y2 + y3) = (x1, y1) + (x2 + x3, y2 + y3) = (x1, y1) + ((x2, y2) + (x3, y3)). Thus, addition is associative.

4. Identity element of addition: The additive identity is (0, 0), since (x, y) + (0, 0) = (x + 0, y + 0) = (x, y) for any (x, y) in R².

5. Inverse elements of addition: The additive inverse of (x, y) is (-x, -y), since (x, y) + (-x, -y) = (x - x, y - y) = (0, 0).

6. Closure under scalar multiplication: c(x, y) = (cx, cy), which is of the same form as the original elements in R². Thus, scalar multiplication is closed.

7. Distributivity of scalar multiplication over vector addition: c((x1, y1) + (x2, y2)) = c(x1 + x2, y1 + y2) = (c(x1 + x2), c(y1 + y2)) = (cx1 + cx2, cy1 + cy2) = (cx1, cy1) + (cx2, cy2) = c(x1, y1) + c(x2, y2). Thus, scalar multiplication is distributive over vector addition.

8. Distributivity of scalar multiplication over scalar addition: (c1 + c2)(x, y) = ((c1 + c2)x, (c1 + c2)y) = (c1x + c2x, c1y + c2y) = c1(x, y) + c2(x, y). Thus, scalar multiplication is distributive over scalar addition.

9. Associativity of scalar multiplication: c1(c2(x, y)) = c1(c2x, c2y) = (c1c2x, c1c2y) = (c1c2)(x, y). Thus, scalar multiplication is associative.

10. Identity element of scalar multiplication: The multiplicative identity is 1, since 1(x, y) = (1x, 1y) = (x, y) for any (x, y) in R².

Since R² with the given operations satisfies all the vector space axioms, it is indeed a vector space.

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Answer:

  a) y = |1/2x -2| +4

  b) y = |1/2x -2| +2

  c) y = |1/2x -3/2| +3

  d) y = |1/2x -5/2| +3

Step-by-step explanation:

You want the equations for the translation of y = |1/2x -2| +3 ...

The transformation of a function required to translate it (right, up) by (h, k) units is ...

  f(x) = f(x -h) +k

For (h, k) = (0, 1), the new function is ...

  y = |1/2x -2| +3 +1

  y = |1/2x -2| +4

For (h, k) = (0, -1), the new function is ...

  y = |1/2x -2| +3 -1

  y = |1/2x -2| +2

For (h, k) = (-1, 0), the new function is ...

  y = |1/2(x -(-1)) -2| +3

  y = |1/2(x+1) -2| +3

  y = |1/2x -3/2| +3

For (h, k) = (1, 0), the new function is ...

  y = |1/2(x -1) -2| +3

  y = |1/2x -5/2| +3

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For the value of x = 4/5 (= 0.8) the matrix a = [tex]\left[\begin{array}{cc}5&x\\-6&-5\end{array}\right][/tex] is equal to its own inverse.

The matrix a is given as,

a = [tex]\left[\begin{array}{cc}5&x\\-6&-5\end{array}\right][/tex]

The value of x is such that a is equal to its inverse, that is,

a = [tex]a^{-1}[/tex] ___(1)

Inverse of an matrix, say a, can be calculated using the formula ,

[tex]a^{-1}\\[/tex] = (adjoint of matrix a) / (determinant of matrix a)

Therefore, Adjoint of matrix a = [tex]\left[\begin{array}{cc}-5&-x\\6&5\end{array}\right][/tex]

where,

As element of the adjoint matrix in row 1 and column 1 is cofactor of the matrix a in row 1 and column 1,  

As element of the adjoint matrix in row 1 and column 2 is cofactor of the matrix a in row 1 and column 2,  

As element of the adjoint matrix in row 2 and column 1 is cofactor of the matrix a in row 2 and column 1,

And  as element of the adjoint matrix in row 2 and column 2 is cofactor of the matrix a in row 2 and column 2.

Therefore, determinant of matrix a = (5)(-5) - (-6)(x) = -25 +30x

Thus from the formula of inverse of a matrix we get,

[tex]a^{-1}\\[/tex] = {1/( -25 +30x)} [tex]\left[\begin{array}{cc}-5&-x\\6&5\end{array}\right][/tex]

=[tex]\left[\begin{array}{cc}-5/ (-25 +30x)&-x/ (-25 +30x)\\6/ (-25 +30x)&5/ (-25 +30x)\end{array}\right][/tex] ___(2)

Therefore, equating equation (1) and (2) we get,

[tex]\left[\begin{array}{cc}-5/ (-25 +30x)&-x/ (-25 +30x)\\6/ (-25 +30x)&5/ (-25 +30x)\end{array}\right][/tex] = [tex]\left[\begin{array}{cc}5&x\\-6&-5\end{array}\right][/tex]

⇒ -5/ (-25 +30x) = 5,

-x/ (-25 +30x) = x,

6/ (-25 +30x) = -6,

and 5/ (-25 +30x) = -5

From any one of the above four equation we can equate for the value of x we get,

5/ (-25 +30x) = -5

⇒1/ (-25 +30x) = -1

⇒ 25 - 30x = 1

⇒ 30x = 24

⇒ x =24/30 = 4/5 (=0.8)

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S cannot be a basis for [tex]R^{n }[/tex]

What is Matrix ?

A matrix is a rectangular array of numbers or symbols arranged in rows and columns. Matrices are commonly used in mathematics, physics, engineering, computer science, and other fields to represent systems of linear equations, transformations, and other mathematical objects and operations.

The statement "For each b in  [tex]R^{m }[/tex], the equation Ax - b has a solution" is equivalent to the following statements:

A. The columns of A span [tex]R^{m }[/tex]

B. Each b in [tex]R^{m }[/tex] is a linear combination of the columns of A.

E. The matrix A has a pivot position in each column.

To explain why S cannot be a basis for  [tex]R^{n }[/tex] , we can use the fact that a set of vectors S = {v1, ..., vk} is a basis for  [tex]R^{n }[/tex] if and only if the matrix whose columns are the vectors in S is invertible. In this case, since k < n, the matrix whose columns are the vectors in S cannot be invertible because it has more columns than rows.

Therefore, S cannot be a basis for [tex]R^{n }[/tex].

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Answer:

a=8c-b/2

Step-by-step explanation:

a/2c+b/4=2

find the lcm

lcm=4c

multiply through by lcm 4c

4c×a/2c+4c ×b/4=4c×2

2×a+b×c=8c

2a+bc=8c

subtract bc from both sides

2a+bc-bc=8c-bc

2a=8c-b

To make 'a' subject of formula, divide 2 from both sides

2a/2=8c-b/2

a=8c-b/2

The population density for each animal is given as follows:

The population density is calculated as the division of the total population by the total area.

The area for this problem is given as follows:

2.22 million acres = 2,220,000 acres.

Hence the densities are given as follows:

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The final equation of the circle is: (x - 4)²+(y - 3)² =8.

The set of all points in a plane that are equally spaced from a fixed point known as the centre is described by the equation of a circle.

It is usually written in the form (x-h)² + (y-k)² = r², where (h, k) represents the center and r represents the radius of the circle.

To find the equation of the circle, we need to find the center of the circle and its radius using the given endpoints of the diameter.

The circle's centre corresponds to the diameter PQ's midpoint.Taking the average of the x-coordinates and the average of the y-coordinates will yield the midpoint's coordinates:

x-coordinate of midpoint = (6 + 2)/2 = 4

y-coordinate of midpoint = (5 + 1)/2 = 3

(4, 3) is the center of circle.

The radius of circle is half the distance between the endpoints of      diameter:

r = 1/2 × √((6 - 2)² + (5 - 1)²) = 1/2 × √(16 + 16) = 1/2 × √(2) = 2√(2)

Therefore, the equation of the circle with center (4, 3) and radius 2√(2) is:

(x - 4)² + (y - 3)² = (2×(2))²

Simplifying and expanding  right-hand side:

(x - 4)² + (y - 3)² = 8

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