(x-6)^2+(y+5)^2=16 In the x y -plane, the graph of the...

$$(x-6)^{2}+(y+5)^{2}=16$$
In the $x y$ -plane, the graph of the equation above is a circle. Point $P$ is on the circle and has coordinates $(10,-5) .$ If $\overline{P Q}$ is a diameter of the circle, what are the coordinates of point $Q$?
A) $(2,-5)$
B) $(6,-1)$
C) $(6,-5)$
D) $(6,-9)$

Labs

Want to see this concept in action?

NEW

Explore this concept interactively to see how it behaves as you change inputs.

View Labs

Key Concepts

Midpoint Formula

The midpoint formula calculates the midpoint of a line segment by averaging the x-coordinates and the y-coordinates of its endpoints. In the context of a circle, the center is the midpoint of any diameter, and this concept is used to find an unknown endpoint when the other endpoint and the center are known.

Diameter of a Circle

A diameter is a chord that passes through the center of the circle, connecting two points on its circumference. It is the longest chord in the circle, and its endpoints are symmetrical about the center, which is a key property when solving problems involving diametrically opposite points.

Standard Equation of a Circle

The standard equation of a circle is written as (x - h)² + (y - k)² = r², where (h, k) is the center and r is the radius. This form is fundamental in coordinate geometry for deriving properties of the circle and for graphing it in the plane.

Center and Radius

Extracting the center and radius from the equation involves identifying the coordinates (h, k) from the shifted terms and computing the radius from the right side of the equation. This information is essential for understanding the circle's position and size in the coordinate system.

Transcript

Please give Ace some feedback