Question

If the line $2x - 3y = k$ touches the parabola $y^2 = 6x$, then find the value of $k$.

• A. -15/4
• B. -27/4
• C. -1/4
• D. -3/4

Answer 1

Answer: (B)

Given $x=\frac{3 y +k}{2} ...(1)$
$y^{2}=6 x ...(2)$
$\Rightarrow y^{2}=6\left(\frac{3 y+ k}{2}\right)$
$\Rightarrow y^{2}=3(3 y+ k)$
$\Rightarrow y^{2}-9 y-3 k=0...(3)$
If line (1) touches parabola (2) then roots of quadratic equation (3) is equal $\therefore (-9)^{2}=4 \times 1 \times(-3 k)$
$\Rightarrow k=-27 / 4$