An equation of the chord of the circle $x^2+y^2=a^2$  passing through the point $\left(2,3\right)$  farthest from the centre is

Let $P\left(2,3\right)$  be the given point, M be the middle point of a chord of the circle $x^2+y^2=a^2$ through P.

Then the distance of the centre O of the circle from the chord is OM.

and ${\left(OM\right)}^2={\left(OP\right)}^2-{\left(PM\right)}^2$

Which is maximum when PM is minimum i.e, M coincides with p i.e. P is the middle point of the chord.

Hence the equation of the chord is $2.x+3.y-a^2={\left(2\right)}^2+{\left(3\right)}^2-a^2$ $\Rightarrow 2x+3y=13$