Find the equation of the locus of a point, the tangents from which to

1.	Find the equation of the locus of a point, the tangents from which to the parabola y2 = 18x are such that sum of their slopes is -3.
Answer» Given equation of the parabola is y² = 18x Comparing this equation with y² = 4ax, we get ⇒ 4a = 18 ⇒ a = 9/2 Equation of tangent to the parabola y² = 4ax having slope m is ⇒ y = mx + a/m ⇒ y = mx + 9/2m ⇒ 2ym = 2xm² + 9 ⇒ 2xm² – 2ym + 9 = 0 The roots m₁ and m₂ of this quadratic equation are the slopes of the tangents. m₁ + m₂ = (-2y)/2x = y/x But, m₁ + m₂ = -3 y/x = -3 y = -3x, which is the required equation of locus

Find the equation of the locus of a point, the tangents from which to the parabola y2 = 18x are such that sum of their slopes is -3.

Answer»

Given equation of the parabola is y² = 18x

Comparing this equation with y² = 4ax, we get

⇒ 4a = 18

⇒ a = 9/2

Equation of tangent to the parabola y² = 4ax having slope m is

⇒ y = mx + a/m

⇒ y = mx + 9/2m

⇒ 2ym = 2xm² + 9

⇒ 2xm² – 2ym + 9 = 0

The roots m₁ and m₂ of this quadratic equation are the slopes of the tangents.

m₁ + m₂ = (-2y)/2x = y/x

But, m₁ + m₂ = -3

y/x = -3

y = -3x, which is the required equation of locus