What is the Standard Equation of the Circle Whose Diameter Has Endpoints -53 and 711?

To find the standard equation of the circle whose diameter has endpoints (-53) and (711), we can follow a series of steps:

Step 1: Find the Center of the Circle

The center ((h, k)) of the circle is the midpoint of the diameter. The midpoint can be calculated using the formula:

(h, k) left(frac{x_1 x_2}{2}, frac{y_1 y_2}{2}right)

Substituting the endpoints:

h frac{-5 7}{2} frac{2}{2} 1

k frac{3 11}{2} frac{14}{2} 7

So the center of the circle is ((1, 7)).

Step 2: Find the Radius of the Circle

The radius (r) is half the length of the diameter. We can find the length of the diameter using the distance formula:

d sqrt{(x_2 - x_1)^2 (y_2 - y_1)^2}

Substituting the endpoints:

d sqrt{(7 - (-5))^2 (11 - 3)^2} sqrt{12^2 8^2} sqrt{144 64} sqrt{208} 4sqrt{13}

Therefore, the radius (r) is:

r frac{d}{2} frac{4sqrt{13}}{2} 2sqrt{13}

Step 3: Write the Standard Equation of the Circle

The standard equation of a circle with center ((h, k)) and radius (r) is given by:

(x - h)^2 (y - k)^2 r^2

Substituting (h 1), (k 7), and (r^2 (2sqrt{13})^2 4 cdot 13 52), we get:

(x - 1)^2 (y - 7)^2 52

Therefore, the standard equation of the circle is:

box(x - 1)^2 (y - 7)^2 52

Alternative Method Using Geometric Properties

Another method to find the circle's equation involves understanding the geometric properties. Any point on the circle joined with the endpoints of any diameter forms a right-angled triangle. Additionally, the product of slopes of perpendicular lines is (-1).

Solving the system:

(frac{y - 3}{x - (-5)} cdot frac{y - 11}{x - 7} -1)

After simplification, we find:

(x - 16x - 1 - 6y - 7 - 4 0)

Simplifying further:

((x - 16)^2 (-6y - 7 - 4)^2 0)

This results in:

((x - 1)^2 - 6^2 (y - 7)^2 - 4^2 0)

Which simplifies to:

((x - 1)^2 (y - 7)^2 52)

General Solution

Given the endpoints (-53) and (711), the center of the circle is ((17, 7)).

The radius of the circle is the distance from the center to one of the endpoints:

(sqrt{(7 - 17)^2 (11 - 7)^2} sqrt{10^2 4^2} sqrt{100 16} sqrt{116} 2sqrt{29})

The standard equation of the circle is:

((x - 17)^2 (y - 7)^2 (2sqrt{29})^2)

Thus, the equation simplifies to:

((x - 17)^2 (y - 7)^2 116)