Question

If the equation $k{y}^2+y=x^2-16x+64$ represents a parabola and $\Delta$ is the discriminant, then the value of $|4\Delta |$ is

Answer 1

The given equation is $k{y}^2+y=x^2-16x+64$
Comparing it with standard conic equation, we get
$a=1,b=-k,h=0,g=-8,f=-1/2,c=64$
For equation to be parabola,
$h^2=ab\Rightarrow k=0$

Now, $\Delta =abc+2fgh-a{f}^2-b{g}^2-c{h}^2$
$=0+0-\frac{1}{4}=-\frac{1}{4}\Rightarrow |4\Delta |=1$