JEE PYQ: Application Of Derivatives Question 17
Question 17 - 2021 (25 Feb Shift 1)
If the curves $\frac{x^2}{a} + \frac{y^2}{b} = 1$ and $\frac{x^2}{c} + \frac{y^2}{d} = 1$ intersect each other at an angle of $90°$, then which of the following relations is true?
(1) $a + b = c + d$
(2) $a - b = c - d$
(3) $ab = \frac{c+d}{a+b}$
(4) $a - c = b + d$
Show Answer
Answer: (2)
Solution
For orthogonal intersection: $\frac{-bx}{ay} \cdot \frac{-dx}{cy} = -1 \Rightarrow bdx^2 = -acy^2$. Subtracting curve equations: $\left(\frac{1}{a} - \frac{1}{c}\right)x^2 + \left(\frac{1}{b} - \frac{1}{d}\right)y^2 = 0$. Using $bdx^2 = -acy^2$: $(c-a) - (d-b) = 0 \Rightarrow a - b = c - d$.
BETA
