Question

The area of the parallelogram formed by the tangents at the ends of conjugate diameters of an ellipse is –

• A. Constant and is equal to the product of the axes
• B. Can not be constant
• C. Constant and is equal to the two lines of the product of the axes
• D. None of these

Answer 1

Answer: (A)

Constant and is equal to the product of the axes

Answer 2

Area of parallelogram

T₁T₂T₃T₄ = 4(Area of parallelogram CPT₂D)

= 4(2 Area of DCPD)

= 8(Area of DCPD)

= 8 × $\frac { 1 } { 2 }$ $\left| \begin{matrix} 0 & 0 & 1 \ a\cos\theta & b\sin\theta & 1 \

• a\sin\theta & b\cos\theta & 1 \end{matrix} \right|$

= 4(ab cos² q + ab sin² q)

= 4ab = 2a × 2b

= Product of the axes of the ellipse.