For 0<θ<(π)/2, four tangents are drawn at the four points ±3cosθ,±2sinθ to the ellipse x^29+y^24=1. If Aθ denotes the area of the quadrilateral formed by these four tangents, the minimum value of Aθ is

Correct answer: (B) 24. We have, Equation of ellipse x^29+y^24=1 Equation of tangent at 3cosθ,2sinθ is x/3cosθ+y/2sinθ=1 Intercept of tangent is 3secθ,0 and 0,2cosecθ Area of quadrilateral =4×1/23secθ×2cosecθ =12secθcosecθ=24/(sin2θ) Minimum area =24

Q31 / 91 · KVPY 2018 (SB/SX) · Mathematics

For $0 < θ < \frac{π}{2}$ , four tangents are drawn at the four points $(\pm 3 \cos θ, \pm 2 \sin θ)$ to the ellipse $\frac{x^{2}}{9} + \frac{y^{2}}{4} = 1 .$ If $A (θ)$ denotes the area of the quadrilateral formed by these four tangents, the minimum value of $A (θ)$ is

A
$21$
Correct answer
B
$24$
C
$27$
D
$30$

Solution

We have, Equation of ellipse

$\frac{x^{2}}{9} + \frac{y^{2}}{4} = 1$

Equation of tangent at $(3 \cos θ, 2 \sin θ)$ is

$\frac{x}{3} \cos θ + \frac{y}{2} \sin θ = 1$

Intercept of tangent is $(3 \sec θ, 0)$ and $(0, 2 cosec θ)$

Area of quadrilateral

$= 4 \times \frac{1}{2} 3 \sec θ \times 2 cosec θ$

$= 12 \sec θ cosec θ = \frac{24}{\sin 2 θ}$

Minimum area $= 24$

Hence, the correct answer is option: 'B'.

For 0<θ<π2, four tangents are drawn at the four points ±3cosθ,±2sinθ to the ellipse x29+y24=1. If Aθ denotes the area of the quadrilateral formed by these four tangents, the minimum value of Aθ is

Solution

More Ellipse Questions

For $0 < θ < \frac{π}{2}$ , four tangents are drawn at the four points $(\pm 3 \cos θ, \pm 2 \sin θ)$ to the ellipse $\frac{x^{2}}{9} + \frac{y^{2}}{4} = 1 .$ If $A (θ)$ denotes the area of the quadrilateral formed by these four tangents, the minimum value of $A (θ)$ is