The co-ordinates of the mid-point of the chord cut off by theline $2 x-5 y+18=0$ by the circle…
The co-ordinates of the mid-point of the chord cut off by theline $2 x-5 y+18=0$
by the circle $x^{2}+y^{2}-6 x+2 y-54=0$ are
$(1,4)$
$(2,4)$
$(4,1)$
$(1,1)$
Solution
Let $A B$ be the chord
Let the mid point of chord be $\mathrm{M}(\mathrm{h}, \mathrm{k})$
Here centre is $\mathrm{O}=(3-1)$
Here $O M \perp A B$
$\therefore($ slope of $\mathrm{OM})($ slope of $\mathrm{AB})=-1$
$\left(\frac{\mathrm{k}+1}{\mathrm{~h}-3}\right)\left(\frac{2}{5}\right)=-1$
$\therefore 5 \mathrm{~h}+2 \mathrm{k}=13$
Point $M(h, k)$ lies on the $2 x-5 y+18=0$...(1)
$\therefore 2 \mathrm{~h}-5 \mathrm{k}=-18$...(2)
Solving equation (1) \& (2) we get $\mathrm{h}=1, \mathrm{k}=4 \Rightarrow(1,4)$ is required point.