Question

A line passing through $P(2, 3)$ and making an angle of $30^\circ$ with the positive direction of $x$-axis meets $x^2 - 2xy - y^2 = 0$ at $A$ and $B$. Then the value of $PA \cdot PB$ is

• A. $17\sqrt{3} + 1$
• B. $17(\sqrt{3} + 1)$
• C. $17(\sqrt{3} - 1)$
• D. $17\sqrt{3} - 1$

Hint:

Substitute parametric line into conic, then use chord length product property for $PA \cdot PB$.

Answer 1

The Correct Option is B

Line through $P(2, 3)$ at $30^\circ$ has slope $m = \tan 30^\circ = \dfrac{1}{\sqrt{3}}$
So equation is: $y - 3 = \dfrac{1}{\sqrt{3}}(x - 2)$
Substitute into the conic $x^2 - 2xy - y^2 = 0$
Solve resulting quadratic in $x$, find product of roots which gives $PA \cdot PB$ (since $P$ lies on the chord, symmetric product)
Final result simplifies to $17(\sqrt{3} + 1)$