Question

If the curves, x²/a + y²/b = 1 and x²/c + y²/d = 1 intersect each other at an angle of 90°, then which of the following relations is TRUE?

• A. a + b = c + d
• B. a - b = c - d
• C. ab / (a+b) = (c+d)/(a+b)
• D. a - c = b + d

Hint:

Two central conics are orthogonal if they are confocal. The condition $a-b = c-d$ ensures the difference of the squares of the axes is constant.

Answer 1

The Correct Option is B

Step 1: Differentiate both curves to find slopes $m_1$ and $m_2$ at intersection $(x_1, y_1)$.
Step 2: $m_1 = -\frac{bx_1}{ay_1}$ and $m_2 = -\frac{dx_1}{cy_1}$. For orthogonality, $m_1m_2 = -1$.
Step 3: $\frac{bdx_1^2}{acy_1^2} = -1 \Rightarrow \frac{x_1^2}{y_1^2} = -\frac{ac}{bd}$.
Step 4: Subtracting curve equations: $x_1^2(\frac{1}{a} - \frac{1}{c}) + y_1^2(\frac{1}{b} - \frac{1}{d}) = 0$.
Step 5: Substituting $x_1^2/y_1^2$ leads to the condition $a - b = c - d$.