length of the tangent to the circle

Question

derive the formula for finding the length of the tangent to the circle (x+5)^2 + (y+9)^2 = 25 from the point (6,8)?

Answer 1

The circle equation (x + 5)² + (y + 9)² = 25

(x - (- 5))² + (y - ( - 9))² = 5²

Compare it to circle equation (x - h)² + (y - k)² = r²

Center of circle = C(- 5, -9)

Radius is r = 5.

From the figure ABC is a right angle triangle.

AB² + BC² = AC²

AB² = AC² - BC²

AB = √(AC² - BC²)

A(x₁, y₁) = (6, 8) and C(x₂,y₂) = (- 5, - 9)

AC = √[(- 5 - 6)² + (- 9 - 8)²]

= √ (121 + 289) = √410

BC = 5

AB = √(410 - 25)

Length of tangent AB = √385 units.