The tangents at the extremities of the latus rectum of the ellipse 3x2+4y2=12 form a rhombus PQRS . Area (in sq. units) of the rhombus PQRS outside the ellipse is equal to

Question

The tangents at the extremities of the latus rectum of the ellipse 3x2+4y2=12 form a rhombus PQRS . Area (in sq. units) of the rhombus PQRS outside the ellipse is equal to (A) 8-2√3π  (B) 12-2√3π  (C) 14-π  (D) 16-2√3π

Tardigrade Admin · Accepted Answer

<img class="img-fluid question-image" alt="Solution" src="https://cdn.tardigrade.in/q/nta/m-lxmplhajo2vlf58b.jpg" /> One of the end of latus rectum is (1 , (3/2)) Tangent at (1 , (3/2)) is x+2y=4 Area of rhombus =4((1/2) ⋅ 4 ⋅ 2)=16 Hence, the required area =16-2√3π sq. units

Q. The tangents at the extremities of the latus rectum of the ellipse $3 x^{2} + 4 y^{2} = 12$ form a rhombus $PQRS$ . Area (in sq. units) of the rhombus $PQRS$ outside the ellipse is equal to

$8 - 23 π$

$12 - 23 π$

$14 - π$

$16 - 23 π$

One of the end of latus rectum is $(1, \frac{3}{2})$
Tangent at $(1, \frac{3}{2})$ is $x + 2 y = 4$
Area of rhombus $= 4 (\frac{1}{2} \cdot 4 \cdot 2) = 16$
Hence, the required area $= 16 - 23 π$ sq. units

Q. The tangents at the extremities of the latus rectum of the ellipse 3x2+4y2=12 form a rhombus PQRS . Area (in sq. units) of the rhombus PQRS outside the ellipse is equal to

8−23​π

12−23​π

14−π

16−23​π