Question

Find the equation of a line which touches the parabola 9x² + 12x + 18y – 14 = 0 and passes through the point (0, 1).

Answer 1

Equation of the given parabola can be written as

9x² + 12x + 4 + 18y - 18 = 0

i.e  (3x + 2)² = -18(y-1)

i.e (x + 2/3)² = -2(y - 1)  ----- (1)

Equation of the tangent to the above parabola can be written as

x + 2/3 = m(y - 1) - 1/2m [∴ 4a = -2 ∴ a = -1/2]  ----(2)

If the tangent passes through (0, 1), then we have

0.m² – 4m – 3 = 0

gives m = – 3/4, ∞

Hence, equation of the required lines, are

x + 2/3 = -3/4(y - 1), x + 2/3 = ∞(y - 1)

i.e. 12x + 9y - 1 = 0 and y - 1 = 0