If the minimum area of the triangle formed by a tangent to the ellipse (x2/b2)+(y2/4 a2)=1 and the co-ordinate axis is kab, then k is equal to

Question

If the minimum area of the triangle formed by a tangent to the ellipse (x2/b2)+(y2/4 a2)=1 and the co-ordinate axis is kab, then k is equal to  (A)  (B)  (C)  (D)

Tardigrade Admin · Accepted Answer

Tangent (x cos θ/b)+(y sin θ/2 a)=1 <img class="img-fluid question-image" alt="image" src="https://cdn.tardigrade.in/img/question/mathematics/f04b99a1a7ff5e13762a41f3e26f3f71-.png" /> So, area (Δ OAB )=(1/2) × ( b / cos θ) × (2 a / sin θ) =(2 a b/ sin 2 θ) ≥ 2 a b ⇒ k=2

Q. If the minimum area of the triangle formed by a tangent to the ellipse $\frac{x ^{2}}{b ^{2}} + \frac{y ^{2}}{4 a ^{2}} = 1$ and the co-ordinate axis is kab, then $k$ is equal to ___

Tangent
$\frac{x c o s θ}{b} + \frac{y s i n θ}{2 a} = 1$

So, area $(Δ O A B) = \frac{1}{2} \times \frac{b}{c o s θ} \times \frac{2 a}{s i n θ}$
$= \frac{2 ab}{s i n 2 θ} \geq 2 ab$
$\Rightarrow k = 2$

Q. If the minimum area of the triangle formed by a tangent to the ellipse b2x2​+4a2y2​=1 and the co-ordinate axis is kab, then k is equal to ___

Tangent bxcosθ​+2aysinθ​=1 So, area (ΔOAB)=21​×cosθb​×sinθ2a​ =sin2θ2ab​≥2ab ⇒k=2

Q. If the minimum area of the triangle formed by a tangent to the ellipse $\frac{x ^{2}}{b ^{2}} + \frac{y ^{2}}{4 a ^{2}} = 1$ and the co-ordinate axis is kab, then $k$ is equal to ___

Tangent
$\frac{x c o s θ}{b} + \frac{y s i n θ}{2 a} = 1$

So, area $(Δ O A B) = \frac{1}{2} \times \frac{b}{c o s θ} \times \frac{2 a}{s i n θ}$
$= \frac{2 ab}{s i n 2 θ} \geq 2 ab$
$\Rightarrow k = 2$