Question

The length of the chord $AB$ of the circle $x^2+y^2-6x+8y-13=0$ whose midpoint is $(2,-3)$ is (units)

Answer 1

Given circle is $x^2+y^2-6x+8y-13=0$
Length of chord with given midpoint
$=2\sqrt{|S_1|}=2\sqrt{|4+9-12-24-13|}$
$=2\sqrt{|-36|}=12$ units


Alternate solution :
Given circle is $x^2+y^2-6x+8y-13=0$
Centre and radius are
$C=(3,-4),r=\sqrt{38}$
Distance between centre and midpoint of the chord is
$d=\sqrt{1^2+1^2}=\sqrt{2}$
Length of chord
$=2\sqrt{r^2-d^2}$
$=2\sqrt{38-2}=12$ units