Circle Word problem

Question

Recall the equation for a circle with center (ℎ,k) and radius r. At what point in the first quadrant does the line with equation y=0.5x+3 intersect the circle with radius 5 and center (0, 3)?

Answer 1

The equation for a circle with the center at (h,k) and the radius r is as below:

( x- h)² + (y - k)² = r²

Since the center is given as (0,3) and the radius as 5, the equation will be:

(x - 0)² + (y - 3)² = 5² ; that is, x² + (y - 3)² = 25

You can find the intersecting points of the circle and the given line

by substituting the line expression (0.5x + 3) for y in the circle equation:

x² + (0.5x +3 -3)² = 25

x² + (0.5x)² = 25

x² + 0.25x² = 25

1.25x² = 25

x² = 25 / 1.25

x² = 20

x = √20(≈4.5) or -√20(≈-4.5)

Since we are interested in the intersection in the first quadrant,

substute only √20 for x in the line equation to find the y coordinate of the intersection:

y = 0.5(√20≈) + 3 ≈ 2.2 + 3 = 5.2

Therefore, the coordinate of the intersection in the first quadrant is about (4.5 , 5.2).

Answer 2

An equation of the circle is (x-0)² + (y-3)² = 25

Replace y by 0,5x + 3 to get x² + (1/4)x² = 25

Solve for x.