Question

Let $7x^2-2y^2+12xy-2x+14y-22=0$ be the hyperbola. Then match the following:

Answer 1

If the point $(x_1,y_1)$ lies outside, on or inside the hyperbola $\frac{x^2}{a^2}-\frac{y^2}{b^2}=1$ then $\frac{x_1^2}{a^2}-\frac{y_1^2}{b^2}-1$ will be $<0$ or $=0$ or $>0$ respectively.

(A) $S_1(0,0)=0-0+0-0+0-22=-22<0$, Outside$\Rightarrow$ option $2$

(B) $S_1(1,0)=7-0+0-2+0-22==7-22=-15<0$, Outside$\Rightarrow$ option $2$

(C) $S_1(1,2)=7-8+24-2+28-22=27>0$, Inside$\Rightarrow$ option $1$

(D) $S_1(0,1)=0-2+0-0+14-22=-10<0$, Outside$\Rightarrow$ option $2$