Question

If $XY=1$ is the new form of the locus $xy-3x+2y-7=0$, when origin is shifted to $A(h,k)$, axes remaining parallel, find values of $h,k$.

Answer 1

Given locus is $xy-3x+2y-7=0$ ..... $\left(i\right)$

Suppose the coordinate of the point $Q$ are $(h,k)$

By the formula for the shift of origin, we have

$x=X+h$ and $y=Y+k$

Putting this values in the eq${}^n$ $\left(i\right)$, we have

$(X+h)(Y+k)-3(X+h)+2(Y+k)-7=0$

The eq${}^n$ is transferred in

$XY=1$

$\Rightarrow XY-1=0$

So here we get $k-3=0$ and $h+2=0$

and $hk-3h+2k-7=-1$

Solving this $k=3,h=-2$

Thus, the co-ordinate is $(h,k)=(-2,3)$.

Hence, the value of $h$ and $k$ are $-2$ and $3$ respectively.