Circles? tangency? an equation of the line passing through the center of the circle?? ?

Question

find the constant value of k so that the line having equation y=-3/4 + k is tangent to the circle whose equation os (x-3)^2 + (y+4)^2 = 25

i need solutions... thanks!

Answer 1

The center of the given circle is (3, -4) and radius is 5.

There seems to be misprint in the equation of the line. I assume it to be

y = -3/4 x + k

=> 3x + 4y - 4k = 0

For the line to be a tangent to the circle, perpendicular distance from the center of the circle should be equal to the radius of the circle.

=> l 3*3 + 4*(-4) - 4k l / 5 = 5

=> l 4k + 7 l = 25

=> 4k + 7 = ±25

=> k = 9/2 or -8

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verification :

For k = 9/2, equation of line is

y = -3/4 x + 9/2

=> 3x + 4y = 18

Perpendicular distance from the center (3, -4) of the circle is

l 3*3 + 4*(-4) - 18 l / 5 = 5 = radius of the circle.

For k = -8, equation of the line is

y = -3/4 x - 8

=> 3x + 4y + 32 = 0

Perpendicular distance from the center (3, -4) of the circle is

l 3*3 + 4*(-4) + 32 l / 5 = 5 = radius of the circle.