Question

Let the tangent drawn to the parabola y2 = 24x at the point (α, β) is perpendicular to the line 2x + 2y = 5. Then the normal to the hyperbola
$\frac{x^2}{α^2}−\frac{y^2}{β^2}=1$
at the point (α + 4, β + 4) does NOT pass through the point

• A. (25, 10)
• B. (20, 12)
• C. (30, 8)
• D. (15, 13)

Answer 1

Answer: (D)

(15, 13)

Answer 2

Any tangent to y 2 = 24 x at (α, β)
β y = 12(x + α)
Slope=$\frac{12}{β}$ and perpendicular to 2x+2y=5
$\frac{12}{β}=1$
β=12,
α=6
Hence, hyperbola is
$\frac{x^2}{36}−\frac{y^2}{144}=1$
and normal is drawn at (10, 16)
Equation of normal
$\frac{36⋅x}{10}+\frac{144⋅y}{16}=$36+144
$=\frac{x}{50}+\frac{y}{20}=1$
This does not pass though (15, 13) out of given option.
So, the correct option is (D): (15, 13)