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?

Solution

Maths sir · 4 steps

Find x₁²/y₁² twice, once from perpendicular slopes and once by subtracting the curves, then equate.

1. Slope of each curve. Let the curves meet at (x₁, y₁). Differentiating x²/a + y²/b = 1 gives m₁ = -(b x₁)/(a y₁). In the same way, the second curve gives m₂ = -(d x₁)/(c y₁).
2. Use the right angle. Perpendicular tangents need m₁ m₂ = -1, so (bd x₁²)/(ac y₁²) = -1. This gives x₁²/y₁² = -ac/bd.
3. Subtract the curves. The point lies on both curves, so subtracting the equations gives x₁²(1/a - 1/c) = y₁²(1/d - 1/b). That is x₁² · (c-a)/ac = y₁² · (b-d)/bd, so x₁²/y₁² = (ac (b-d))/(bd (c-a)).

   (ac (b-d))/(bd (c-a)) = -ac/bd ⇒ b - d = a - c

4. So. Rearranging b - d = a - c gives a - b = c - d, which is option (d).

The idea to keep. When two conics cut at right angles, get x²/y² at the meeting point in two ways: from m₁ m₂ = -1 and from subtracting the curve equations. Equating the two gives the condition.

Answer from the NTA final answer key. Solution by PrepWiser.