Question

If two chords $ABandAC$ drawn from the point $A(4,4)$ to the parabola $x^2=4y$ are divided by line $y=mx$ in the ratio $2:1$ then

• A. $m<-\sqrt{3}$
• B. $m<-\sqrt{3}-1$
• C. $m>-1$
• D. $m>\sqrt{3}-1$

Answer 1

Answer: (B), (D)

$m<-\sqrt{3}-1$
$m>\sqrt{3}-1$

Point $P(4,4)$ lies on the parabola.
Let $Q(x_1,y_1)$ be a point on the parabola.
Let the point of intersection of the line $y=mx$ with the chords be $(a,ma)$, then
$a=\frac{4+2x_1}{3}$
$\Rightarrow x_1=\frac{3a-4}{2}$
and $ma=\frac{4+2y_1}{3}$
$\Rightarrow y_1=\frac{3ma-4}{2}$
As $(x_1,y_1)$ lies on the curve
$\Rightarrow (3a-4{)}^2=8(3ma-4)$
$\Rightarrow 3a^2-8(1+m)a+16=0$
For two distinct chords, $D>0$
$\Rightarrow (1+m{)}^2-3>0$
$\Rightarrow (m+1+\sqrt{3})(m+1-\sqrt{3})>0$
$\Rightarrow m>\sqrt{3}-1$ or $m<-\sqrt{3}-1$