equation pf circle

(a) L : x + 3y - 9 = 0 => x = -9 - 3y

Sub. into C: x^2 + y^2 - 2x - 9 = 0

(-9 - 3y)^2 + y^2 - 2(-9 - 3y) - 9 = 0

9y^2 + 54y + 81 + y^2 + 18 + 6y - 9 = 0

10y^2 + 60y + 90 = 0

Discriminant = 0 and so L1 is a tangent to C

(b)(i) The slope of L2 is 3

(ii) Let L2: y = 3x + c and sub. into C

x^2 + y^2 - 2x - 9 = 0

x^2 + (3x + c)^2 - 2x - 9 = 0

10x^2 + (6c - 2)x + (c^2 - 9) = 0...(*)

Discriminant = 0

36c^2 - 24c + 4 - 40c^2 + 360 = 0

-4c^2 - 24c + 364 = 0

c^2 + 6c - 91 = 0

(c + 13)(c - 7) = 0

c = 7 or -13

When c = 7, * becomes x^2 + 4x + 4 = 0 => contact point (-2,1)

L2: y = 3x + 7

When c = -13, * becomes x^2 - 8x + 16 = 0 => contact point (4,-1)

L2: y = 3x - 13