Question

Let $y=mx+c$ is a tangent at point $P$ on the curve $xy-2x^2=0,$ meets the curve again at $Q.$ Then which of the following(s) is(are) not correct

• A. $c=0,m=3$
• B. $c=1,m=-1$
• C. $c=2,m=2$
• D. $c=-2,m=2$

Answer 1

Answer: (A), (C), (D)

$c=-2,m=2$

As, $y=mx+c$ is tangent at $P$ and intersects the curve at $Q.$
Now, solving the equation of tangent with the equation of curve.
$\Rightarrow x(mx+c)-2x^2=0$
$\Rightarrow (m-2)x^2+cx=0$
$\Rightarrow x\left(\right(m-2)x+c)$
$\Rightarrow x=0,\frac{-c}{m-2}$
As, tangent intersects the curve at two distinct points.
$\therefore \frac{-c}{m-2}\ne 0(x=0\text{is already a solution})$ and $m-2\ne 0$
$\Rightarrow c\ne 0$ and $m\ne 2$