The Student Room Group
Fenchurch
A circle has the equation x2 + y2 − 6x + 2y − 15 = 0.

a) Find the coordinates of the centre of the circle.
b) Find the radius of the circle.
c) Show that the tangent to the circle at the point (7, 2) has equation
4x + 3y − 34 = 0.

a.) x^2 - 6x + y^2 + 2y = 15
=> (x - 3)^2 - 9 + (y + 1)^2 - 1 = 15
=> (x - 3)^2 + (y + 1)^2 = 25
=> Centre: (3, -1)

b.) Radius^2 = 25 => Radius = 5
c.) x^2 + y^2 − 6x + 2y − 15 = 0
=> 2x + 2y.(dy/dx) - 6 + 2.(dy/dx) = 0
=> 2(dy/dx)[y + 1] = 6 - 2x
=> dy/dx = (3 - x)/(y + 1)
At (7, 2): dy/dx = (-4)/(3) = -4/3
=> Tangent at (7, 2) has gradient -4/3.

Equation of Tangent: y - 2 = -(4/3)(x - 7)
=> y = -4x/3 + 28/3 + 2
=> y = -4x/3 + 34/3
=> 3y = 34 - 4x
=> 4x + 3y - 34 = 0