Question

If $y-mx+c=0,$ $m,c\in R$ is a tangent at point $A$ to the curve $y^2=2x^2-2x,$ meeting the curve again at point $B,$ then which of the following conditions holds TRUE?

• A. $2mc<1+2c^2$
• B. $2mc<1+c^2$
• C. $3mc<1+2c^2$
• D. $mc<1$

Answer 1

The correct option is A $2mc<1+2c^2$

$y=mx-c$ is tangent at $A$ and intersects the curve at $B.$
Solving the equation of tangent with the equation of curve,
$(mx-c{)}^2=2x^2-2x$
$\Rightarrow m^2{x}^2+c^2-2mcx=2x^2-2x$
$\Rightarrow (m^2-2)x^2-2(mc-1)x+c^2=0$
Above equation must have two real and distinct roots.
$\therefore D>0$
$\Rightarrow 4(mc-1{)}^2-4(m^2-2)c^2>0$
$\Rightarrow 4m^2{c}^2+4-8mc-4m^2{c}^2+8c^2>0$
$\Rightarrow 2c^2-2mc+1>0$
$\Rightarrow 2mc<1+2c^2$