Question

The equation of the parabola having focus at $(-1,-2)$ and the directrix $x-2y+3=0$ is $a{x}^2+bxy+c{y}^2+gx+fy+16=0$. If $\alpha$ and $\beta$ are the roots of the equation $g{x}^2-fx+60=0$, then the value of ${\alpha }^3+{\beta }^3$ is

• A. $2(4ab-5c)$
• B. $3(4ab-5c)$
• C. $3(5ab-4c)$
• D. $2(5ab-4c)$

Answer 1

The correct option is D $2(5ab-4c)$

Given, Focus $\equiv (-1,-2)$
and Directrix is $x-2y+3=0$
By definition of the parabola, we know
$\sqrt{(x+1{)}^2+(y+2{)}^2}=\left|\frac{x-2y+3}{\sqrt{5}}\right|$
$\Rightarrow 5(x^2+2x+1+y^2+4y+4)=(x-2y+3{)}^2$
$\Rightarrow 4x^2+4xy+y^2+4x+32y+16=0$

So, we have $a=4,b=4,c=1,g=4,f=32$

Now, $\alpha ,\beta$ are the roots of $4x^2-32x+60=0$
$\Rightarrow x^2-8x+15=0$
$\Rightarrow \alpha =3,\beta =5$ or $\alpha =5,\beta =3$
${\alpha }^3+{\beta }^3=27+125=152$
$2(5ab-4c)=2(80-4)=152$