Hyperbola
Tangent to Hyperbola with Given Slope — Axis Intercepts
nta_pyq_2023_apr
Grade 11
Question:
The foci of a hyperbola are $(\pm2,0)$ and its eccentricity is $\dfrac{3}{2}$. A tangent perpendicular to $2x+3y=6$ is drawn at a point in the first quadrant. If $a$ and $b$ are the $x$- and $y$-intercepts of the tangent, then $|6a|+|5b|$ is equal to
Step-by-Step Solution
Key Concept: $ae=2,\ e=\frac{3}{2}\Rightarrow a=\frac{4}{3}$. $b^2=a^2e^2-a^2=\frac{20}{9}$. Tangent slope $=\frac{3}{2}$ (perp to $2x+3y=6$). Use $y=\frac{3}{2}x\pm\sqrt{a^2\cdot\frac{9}{4}-b^2}$.
Tangent: $y=\frac{3}{2}x-\frac{4}{3}$. $|6\cdot\frac{8}{9}|+|5\cdot(-\frac{4}{3})|=12$.
Correct Answer: 12