Question

The co-ordinates of the middle point of the chord cut-off by $2x-5y+18=0$ by the circle $x^2+y^2-6x+2y-54=0$ are

• A. $(1,4)$
• B. $(2,4)$
• C. $(4,1)$
• D. $(1,1)$

Answer 1

The correct option is A $(1,4)$

Solving $2x-5y+18=0$ .......$\left(1\right)$

$x^2+y^2-6x+2y-54=0$ ........$\left(2\right)$

we should get two points where the chord intersect the circle.

Put $2x=5y-18$ in the equation-$\left(2\right)$ from equation-$\left(1\right)$

we get

$\frac{(5y-18{)}^2}{4}+y^2-3(5y-18)+2y-54=0$

or $\frac{25y^2+324-180y}{4}+y^2-15y+54+2y-54=0$

or $29y^2+324-180y-60y+216+8y-216=0$

or $29y^2-232y+324=0$

$y^2-8y+\frac{324}{29}=0$

Compare with $a{y}^2+by+c=0$

sum of $y_1+y_2=-\frac{-8}{1}=8$

$\Rightarrow \frac{y_1+y_2}{2}=4$

So, $y-$coordinate of midpoint is $4$

as $x=\frac{5y-18}{2}$

$\Rightarrow x_1+x_2=\frac{5y_1-18}{2}+\frac{5y_2-18}{2}$

or $x_1+x_2=\frac{5(y_1+y_2)-36}{2}=\frac{5\times 8-36}{2}=2$

$\Rightarrow \frac{x_1+x_2}{2}=1$
So, $x-$coordinate of midpoint is $1$

So mid point $=(1,4)$.