Question

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

• A. $2x+3y=13$
• B. $3x-y=3$
• C. $x-2y+4=0$
• D. $x-y+1=0$

Answer 1

The correct option is A $2x+3y=13$

Let $P(2,3)$ 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 $(OM{)}^2=(OP{)}^2-(PM{)}^2$ $($ From figure $)$

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

$2\cdot x+3\cdot y-a^2=2^2+3^2-a^2$

$\Rightarrow 2x+3y=13.$