Find an equation of the circles:(there are 4 questions)

Question

c)Center on line x=3, tangent y-axis at (0, 5)

d)With center (–2, 0); passes through (2, 0)

e)Radius 5, tangent to y-axis; contain (10,7)

f)Center on line y–4=0, tangent x-axis at (–2, 0)

PLEASE HELP AND THANK YOU IF YOU DO!!!

Answer 1

The general equation of a circle is (x - h)² + (y - k)² = r² ,

where h is the horizontal shift, k is the vertical shift, and r is the radius of the circle.

For example, if we have (x - 2)² + (y - 2)² = 9, then we have a circle of radius 3 at the point (2,2).

c) Center on line x=3, tangent y-axis at (0, 5):

We know that the center is on the point (3,5), because we are given that the center is on the line x = 3, and we only intersect the line x=0 (y axis) at the point (0,5). So, the radius must be r = 3, making the equation of the line:

(x - 3)² + (y - 5)² = 9

d) Center (–2, 0); passes through (2, 0):

A circle can only pass through a point once, so if we know the center and a point on the circle we can find the radius. Because the distance between the center (-2,0) and the point (2,0) is 4, r = 4

(x + 2)² + y² = 16

e) Radius 5, tangent to y-axis; contain (10,7)

We know that center lies on x = 5, because we know that the circle is in Quadrant I from the contained point. The contained point is (10, 7), and the radius is 5, so we have to arrange the center such that it is 5 units away from the point (10, 7). The question is sorted very nicely because the x coordinate of the contained point is exactly one radius away from the x coordinate of the center. We therefore know that the y coordinate of the center is at y = 7. If they gave us some other point other than the farthest possible from x= 5 for a circle of r = 5 we would have 2 options which is important to remember. As it stands,

(x - 5)² + (y - 7)² = 25

f) Center on line y–4=0, tangent x-axis at (–2, 0)

This is the same as (c). The center is on the line y = 4, so the y coordinate is 4. We are tangent to the x axis at (-2,0), so the center is at (-2,4). The radius is the distance from (-2,4) to (-2,0), which is 4.

(x + 2)² + (y - 4)² = 16

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