Q86 / 155 · WBJEE 2025 · Mathematics
The line $y-\sqrt{3} x+3=0$ cuts the parabola $y^2=x+2$ at the points $P$ and $Q$. If the co-ordinates of the point $X$ are $(\sqrt{3}, 0)$, then the value of $X P \cdot X Q$ is
- Correct answerA$\frac{4(2+\sqrt{3})}{3}$
- B$\frac{4(2-\sqrt{3})}{2}$
- C$\frac{5(2+\sqrt{3})}{3}$
- D$\frac{5(2-\sqrt{3})}{3}$
Solution
Arbitrary point on the line $\left(\sqrt{3}+\frac{r}{2}, \frac{r \sqrt{3}}{2}\right)$ satisfying in the parabola $\Rightarrow 3 r^2-2 r-(4 \sqrt{3}+8)=0$
$\because$ roots $r_1$ & $r_2$
$\therefore$ product $r_1 r_2=-\frac{4 \sqrt{3}+8}{3}$
$\therefore X P \cdot X Q=\left|r_1 r_2\right|=\frac{4}{3}(\sqrt{3}+2)$
Hence, the correct answer is option: 'A'.