Conic Sections - Circles

Question

Use the given information to find the equation of the circles.

center at (-2, 8) and passes through (1,4).

Answer 1

To write the equation of a circle, use the fact that a circle is made of all points that are the same distance from its center. Here the center is at ((-2, 8)), and the circle passes through ((1, 4)), so the radius is the distance between those two points. The horizontal change from (-2) to (1) is (3), and the vertical change from (8) to (4) is (-4). Squaring and adding gives the radius squared: (3^2 + (-4)^2 = 9 + 16 = 25). Using the standard circle form “(x minus the x-coordinate of the center) squared plus (y minus the y-coordinate of the center) squared equals radius squared,” the equation is ((x+2)^2 + (y-8)^2 = 25).

Answer 2

The general equation for a circle centered at (x₀, y₀) with radius r is given by

( x - x₀ )² + ( y - y₀ )² = r² (1)

and we are given the center is at (-2,8). So plugging these into (1) we get

( x - (-2) )² + ( y - 8 )² = r² or ( x + 2 )² + ( y - 8 )² = r² (2)

We are also given a point on the circle, (1,4). Plugging this into (2) lets us find r².

( 1 + 2 )² + ( 4 - 8 )² = r² giving r² = 3² + (-4)² = 9 + 16 = 25.

Substituting this back into (2) gives

( x + 2 )² + ( y - 8 )² = 25