Question

Let $P \left( x _ { 1 } , y _ { 1 } \right)$ and $Q \left( x _ { 2 } , y _ { 2 } \right)$are two points such that their abscissa $x _ { 1 }$ and $x _ { 2 }$ are the roots of the equation $x ^ { 2 } + 2 x - 3 = 0$ while the ordinates $y _ { 1 }$ and $y _ { 2 }$ are the roots of the equation $y ^ { 2 } + 4 y - 12 = 0$. The centre of the circle with PQ as diameter is.

• A. $( - 1 , - 2 )$
• B. $( 1,2 )$
• C. $( 1 , - 2 )$
• D. $( - 1,2 )$

Answer 1

Answer: (A)

$( - 1 , - 2 )$

Answer 2

$x _ { 1 } , x _ { 2 }$are roots of $x ^ { 2 } + 2 x + 3 = 0$

⇒ $x _ { 1 } + x _ { 2 } = - 2$

∴ $\frac { x _ { 1 } + x _ { 2 } } { 2 } = - 1$

$y _ { 1 } , y _ { 2 }$ are roots of $y ^ { 2 } + 4 y - 12 = 0$

⇒ $y _ { 1 } + y _ { 2 } = - 4 \Rightarrow \frac { y _ { 1 } + y _ { 2 } } { 2 } = - 2$

Centre of circle $\left( \frac { x _ { 1 } + x _ { 2 } } { 2 } , \frac { y _ { 1 } + y _ { 2 } } { 2 } \right) = ( - 1 , - 2 )$.