Question

The coordinates of middle point of the chord cut off on the line $2x-5y+18=0$ by the circle $x^2+y^2-6x+2y-54=0$ is

• A. $(-4,1)$
• B. $(1,4)$
• C. $(6,6)$
• D. $(-9,0)$

Answer 1

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

Given equation of circle $x^2+y^2-6x+2y-54=0$ compare with $x^2+y^2+2hx+2ky-c$, we get $h=-3,k=1$
Also, the centre of the circle is given by $(-h,-k)=(3,-1)$
Let $M(a,b)$ be the midpoint of the chord, then the point M lies on $2x-5y+18=0$......(1)
$2a-5b=-18$......$\left(2\right)$
$\Rightarrow y=\frac{2x}{5}+\frac{18}{5}$
Slope, $m=\frac{2}{5}$
As the chord and segment joining centre and midpoint are perpendicular.
$\left(\frac{b+1}{a-3}\right)\left(\frac{2}{5}\right)=-1$
$\Rightarrow 5a+2b=13$......$\left(3\right)$
Solving $\left(2\right)$ and $\left(2\right)$:
From (eq(1)$\times$2)$+(5\times$eq(2)),
$\Rightarrow 4a-10b+25a+10b=-36+65$
$\Rightarrow 29a=29$
$\therefore a=1$
Substitute $a=1$ in equation(2), we get
$\Rightarrow 2\left(1\right)-5b=-18$
$\Rightarrow 5b=18+2$
$\Rightarrow 5b=20$
$\therefore b=4$
Hence, the mid point, M($a=1,b=4$)