Question

the minimum area of triange formed by any tangent to the ellipse x²/16+y²/81=1 and the coordinate-axes is​

Answer 1

TOPIC :- QUADRATIC EQUATION

SUB-TOPIC :- TANGENT TO AN ELLIPSE

CONCEPT USED :- Tangent to the ellipse x²/a² + y²/b² = 1 at any point (acosm, bsinm) is x/acosm + y/bsinm = 1.

STEPS

For the ellipse x²/16 + y²/81 = 1, a = 4, b = 9

Thus, tangent :- x/4 cosm + y/9 sinm

Now, this tangent intersect the coordinate axes at (4 cosm, 0) and (0, 9sinm)

Thus, area formed = $\frac{1}{2} \times 4 \cos(m) \times 9 \sin(m \\ ) = >9 \sin(2m)$

Thus, the minimum area is 0. and the maximum area is 9 .

Regards

KSHITIJ