Question

If $2p^2-3q^2+4pq-p=0$ and a variable line px+qy=1 always touches a parabola whose axis is parallel to X-axis, then equation of the parabola is

• A. $(y-4{)}^2=24(x-2)$
• B. $(y-3{)}^2=12(x-1)$
• C. $(y-4{)}^2=12(x-2)$
• D. $(y-2{)}^2=24(x-4)$

Answer 1

The correct option is C $(y-4{)}^2=12(x-2)$

The parabola be $(y-a{)}^2=4b(x-c)$
Equation of tangent is $(y-a)=-\frac{p}{q}(x-c)-\frac{bq}{p}$
Comparing with px+qy=1, we get $c{p}^2-b{q}^2+apq-p=0$
$\therefore \frac{c}{2}=\frac{b}{3}=\frac{a}{4}=1\Rightarrow$ the equation is $(y-4{)}^2=12(x-2)$.