Question

If the parabola $x^2=ay$ makes an intercept of length $\sqrt{40}$ unit on the line $y-2x=1$ then $a$ is equal to

• A. $1$
• B. $-2$
• C. $-1$
• D. $2$

Answer 1

Answer: (A), (B)

The correct options are
A $1$
B $-2$

We know that,
$PQ=\sqrt{40}\Rightarrow (x_1-x_2{)}^2+(y_1-y_2{)}^2=40$
We know that poingts $P,Q$ lies on the line $y=2x+1$,
$\Rightarrow (x_1-x_2{)}^2+(1+2x_1-1-2x_2{)}^2=40\Rightarrow 5(x_1-x_2{)}^2=40\Rightarrow (x_1-x_2{)}^2=8$
Putting $y=1+2x$ in th equation of the parabola,
$x^2=a+2ax\Rightarrow x^2-2ax-a=0$
So the roots of the above quadratic equation will be $x_1,x_2$
$(x_1-x_2{)}^2=(x_1+x_2{)}^2-4x_1{x}_2\Rightarrow 8=(2a{)}^2+4\Rightarrow a^2+a-2=0\Rightarrow (a+2\left)\right(a-1)=0\Rightarrow a=1,-2$