Question

If the parabola $x^2=ay$ makes an intercept of length $\sqrt{40}$ 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 $-2$

Solving the two equations $x^2=ay$ and $y-2x=1$, we get

$x^2=a(2x+1)\Rightarrow x^2-2ax-a=0$

$\therefore x_1+x_2=2a,x_1{x}_2=-a$

So, the given line cuts the parabola at two points $(x_1,y_1)$ and $(x_2,y_2)$.

Now, ${\left(\sqrt{40}\right)}^2={(x_1-x_2)}^2+{(y_1-y_2)}^2$ (Given)

$\Rightarrow 40={(x_1-x_2)}^2+(2x_1+1-2x_2-1{)}^2$

$\Rightarrow 40=5(x_1-x_2{)}^2$

$\Rightarrow (x_1-x_2{)}^2=8$

$\Rightarrow (x_1+x_2{)}^2-4x_1{x}_2=8$

$\Rightarrow 4a^2+4a=8$

$\Rightarrow a^2+a-2=0\Rightarrow (a+2)(a-1)=0\Rightarrow a=1,-2$