Find an equation for the circle whose center is (5, .5) and is tangent to the line whose equation is 5x-12y=20

Question

Please also include the step by step process on how to solve it. Thanks!

EDIT: Should be (5, -5).

Answer 1

You know the form of the equation of the circle is
(x-a)²+(y-b)² = r²

where r is the radius of the circle
(a,b) is the coordinate of the center

So far, you have

(x-5)²+(y+5)² = r² eqn1

For a line with equation Ax + By + C = 0 and a point (r,s) the distance from the point to the line is given by abs(Ar + Bs + C)/sqrt(A² + B²)
This distance will be r. Then you can plug r into eqn1

r = abs[5(5) + (-12)(-5) + (-20)]/sqrt(5²+(-12)²) = abs[65] / sqrt(169) = 65/13 = 5

Your equation becomes
(x-5)²+(y+5)² = 25

Answer 2

Find an equation for the circle whose center is (5, -5) and is tangent to the line whose equation is 5x - 12y = 20.

The slope of the given line is m = -5/-12 = 5/12.

The distance from the center of the circle to the given line is the radius of the circle. First set the equation of the given line equal to zero and then use the distance formula.

5x - 12y = 20
5x - 12y - 20 = 0

The distance from the point (5, -5) to the given line is:

d = | 5*5 - 12*(-5) - 20 | / √[5² + (-12)²]
d = | 25 + 60 - 20 | / √(25 + 144)
d = 65 / √169 = 65/13 = 5

The radius of the circle is 5 and the center is (5, -5). The equation of the circle is:

(x - h)² + (y - k)² = r²

(x - 5)² + (y + 5)² = 25

Answer 3

yet another attitude (in case you do no longer happen to undergo in strategies the gap formula for a factor to a line): tangent is 5x + 12y = thirteen ==> y = -5x/12 + thirteen/12 slope of tangent is -5/12; as a effect slope of radius as much as now would be 12/5 (perpendicular to the tangent) we would like the line that passes by way of (3 , 2) with slope 12/5: y - 2 = 12/5 (x - 3) ==> y = 12/5 x - 36/5 + 10/5 y = 12/5 x - 26/5 <== radius and the factor of intersection between the radius and tangent would be: 12/5 x - 26/5 = -5/12 x + thirteen/12 multiply by 60 to sparkling the fractions 144x - 312 = -25x + sixty 5 169x = 377==> x = 377 / 169 y = 12/5 (377/169) - 26/5 = 2/thirteen distance from (3,2) to (377/169 , 2/thirteen) = 2 (using the gap formula) as a effect, the radius is two, and the circle is given by (x - 3)^2 + (y - 2)^2 = 2^2 = 4 constantly extra effective than one thank you to pores and skin a math cat!

Answer 4

The distance between the given point and the line is the radius of the circle. The line passing through the given point and the point of tangency is perpendicular to the given line (the product of their slopes is -1). Intersection of both lines is the point of tangency.

Find slope of given line.

Find equation of perpendicular line (known slope and point).

Solve for intersection of lines. This is the point of tangency.

Find distance between given point and point of tangency. This is the radius of the circle.

Find equation of circle (known center and radius).