Let the tangent drawn to the parabola y 2=24 x at the point (α, β) is perpendicular to the line 2 x+2 y=5. Then the normal to the hyperbola (x2/α2)-(y2/β2)=1 at the point (α+4, β+4) does NOT pass through the point :

Question

Let the tangent drawn to the parabola y 2=24 x at the point (α, β) is perpendicular to the line 2 x+2 y=5. Then the normal to the hyperbola (x2/α2)-(y2/β2)=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)

Tardigrade Admin · Accepted Answer

Tangent at (α, β) has slope 1 β2=24 α Equation of tangent y β=12( x +α), (12/β)=1 ⇒ α=6, β=12 ∴(α+4, β+4)=(10,16) Normal at (10,16) to (x2/36)-(y2/144)=1 is 2 x+5 y=100

Q. Let the tangent drawn to the parabola $y^{2} = 24 x$ at the point $(α, β)$ is perpendicular to the line $2 x + 2 y = 5$ . 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 :

$(25, 10)$

$(20, 12)$

$(30, 8)$

$(15, 13)$

Tangent at $(α, β)$ has slope 1
$β^{2} = 24 α$
Equation of tangent $y β = 12 (x + α), \frac{12}{β} = 1$
$\Rightarrow α = 6, β = 12$
$∴ (α + 4, β + 4) = (10, 16)$
Normal at $(10, 16)$ to $\frac{x ^{2}}{36} - \frac{y ^{2}}{144} = 1$ is
$2 x + 5 y = 100$

Q. 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 α2x2​−β2y2​=1 at the point (α+4,β+4) does NOT pass through the point :

(25,10)

(20,12)

(30,8)

(15,13)