Circle

Circle C has the equation (x-4)² + (y-8)² =289
The point P (x,y) lies on C.
a) Find, in terms of x and y, the gradient of the tangent of C at P.
b) Hence, or otherwise, find an equation of the tangent to C at the point (21,8).

Centre of circle = (4, 8)

Slope of line joining centre (4,8) and point P(x,y) on C

m = (y – 8)/(x – 4)

Slope of tangent =  – 1/m =  – (x – 4)/(y – 8) = (4 – x)/(y – 8)

Slope of tangent at point (21,8) = (4 – 21)/(8 – 8) = infinity

So, the tangent is parallel to Y axis and equation of tangent is x = 21