find the equation of the line tangent to the circle (x-1)^2+(y-1)^2=25 at the point (4,-3)

Question

It is really hard.

Answer 1

The circle is (x - 1)^2 + (y - 1)^2 = 25 and the point is (4, - 3).

Equation of tangent line : y - y₁ = m(x - x₁).

center of circle is  (h, k ) = (1, 1).

The tangent line is perpendecular to the segment joining (4, - 3) and (1, 1).

Slope of the segment joining (4, - 3) and (1, 1) = 1 + 3/1 - 4 = 4/- 3 = - 4/3.

Because the slopes of perpendicular lines are negative reciprocals, the slope of perpendicular line through (4, - 3) is 3/4.

Substitute the values of m = 3/4 and (x₁, y₁) = (4, - 3) in equation of tangent line.

y - (- 3) = (3/4)(x - 4)

y + 3 = (3/4)x - 3

y = (3/4)x - 3 - 3

y = (3/4)x - 6.

Therefore, the equation of the line tangent to the given circle is y = (3/4)x - 6.