Question

If the equation $p{x}^2+(2-q)xy+3y^2-6qx+30y+6q=0$ represent a circle, then the values of $p$ and $q$ are

• A. $2,2$
• B. $3,1$
• C. $3,2$
• D. $3,4$

Answer 1

Answer: (C)

$3,2$

As we know that if the equation $a{x}^2+b{y}^2+2hxy+2gx+2fy+c=0$ represents a circle then-

1. coefficient of $x^2=$Coefficient of $y^2$, i.e., $a=b$
2. Coefficient of $xy=0$, i.e., $h=0$

Now if the equation $p{x}^2+(2-q)xy+3y^2-6qx+30y+6q=0$ represents the equation of circle then-

• Coefficient of $x^2=$Coefficient of $y^2$

$\Rightarrow p=3$

• Coefficient of $xy=0$
  

$2-q=0\Rightarrow q=2$

Hence if the equation $p{x}^2+(2-q)xy+3y^2-6qx+30y+6q=0$ represents a circle then the value of $p$ and $q$ are $3$ and $2$ respectively.