Question

The origin is shifted to $(1,2)$ the equation$y^2-8x-4y+12=0$ changes to $Y^2+4aX=0$, then $a$ is equal to ?

Answer 1

Since the origin is shifted to $(1,2)$. So, the coordinates of the point (x, y) change to $(X,Y)$.

Substitute$x=X+1,y=Y+2$ in the given equation $y^2-8x-4y+12=0$.

$$\begin{gathered} \therefore {(Y+2)}^2-8(X+1)-4(Y+2)+12=0 \\ \Rightarrow Y^2+4+4Y-8X-8-4Y-8+12=0 \\ \Rightarrow Y^2-8X=0 \end{gathered}$$

Compare the above equation with $Y^2+4aX=0$.

$$\begin{gathered} \therefore 4a=-8 \\ \Rightarrow a=-\frac{8}{4} \\ \Rightarrow a=-2 \end{gathered}$$

Hence, the value of $a$ is equal to $-2$.