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