Length of the chord of contact of parabola

Question

Length of the chord of contact of (2,5) with respect to parabola y²= 8x.

Answer 1

Chord of contact of (2,5) with respect to y² = 8x is

y⋅5 = 4(x + 2)

⇒ 5y − 4x = 8 ⋯(1)

Solving equation (1) with y² = 8x

y² = (5y−8)2

⇒ y² − 10y + 16 = 0

⇒ (y−8) (y−2) = 0

∴ y₁ = 8, y₂ = 2

Using (1), we get

x₁ = 8, x₂ = 12

Then the end points of chord are (8,8) and (12,2)

So, length of chord of contact

\(= \sqrt{(8 - \frac{1}{2})^2 + (8 - 2)^2}\)

\(= \frac{3\sqrt{41}}{2} units\)