Question

State whether given statement is true or false

Prove that the circles $2x^2+2y^2-3x-4y+2=0$ and $x^2+y^2-4x+2y=0$ cut orthogonally.

• A. True
• B. False

Answer 1

The correct option is A True

$C_1:2x^2+2y^2-3x-y+2=0$

OR $x^2+y^2-\frac{3}{2}x-2y+1=0(g=\frac{8}{4}f=1)$

$C_2:x^2+y^2-4x+2y=0$ $(g^{\prime }=2,f^{\prime }=-1)$

Will cut orthogonally if

$2g{g}^{\prime }+2f{f}^{\prime }=C+C^{\prime }$

$LHS,2g{g}^{\prime }+2f{f}^{\prime }=2\times \frac{3}{4}\times \frac{4}{2}-2\times 1\times 1=1$

$RHS\to C+C^{\prime }=1+0=1$

$LHS=RHS$

Hence, circles cut orthogonally.