Hyperbola
Nearest Point on Hyperbola to a Line
nta_pyq_2023_jan
Grade None
Question:
Let $P(x_0, y_0)$ be the point on the hyperbola $3x^2 - 4y^2 = 36$, which is nearest to the line $3x + 2y = 1$. Then $\sqrt{2}(y_0 - x_0)$ is equal to:
Step-by-Step Solution
Key Concept: At the nearest point, the tangent to the hyperbola is parallel to the given line; use slope condition.
Hyperbola: $\frac{x^2}{12}-\frac{y^2}{9}=1$. Tangent slope $=-3/2$. Parametric tangent slope gives point $(x_0,y_0)$; compute $\sqrt{2}(y_0-x_0)=-9$.
Correct Answer: 3