Question

The line $4x+3y-4=0$ divides the circumference of the circle centred at $(5,3)$ in the ratio $1:2$. Then the equation of the circle is

• A. $x^2+y^2-10x-6y-66=0$
• B. $x^2+y^2-10x-6y+100=0$
• C. $x^2+y^2-10x-6y+66=0$
• D. $noneofthese$

Answer 1

The correct option is A $x^2+y^2-10x-6y-66=0$

The line $4x+3y–4$ is a chord and it divides the circumference in $1:2$ ratio

$⟹x+2x=360$

$⟹x=120$

The chord subtends ${120}^{\circ }$ at the center.

Distance of center from chord.

$d=\frac{|20+9-4|}{\sqrt{4^2+3^2}}=5$

According to the diagram

$\cos 60=\frac{d}{r}$

$r=\frac{5}{\frac{1}{2}}=10cm$

Equation of circle will be

$(x-5{)}^2+(y–3{)}^2={10}^2$

$⟹x^2+y^2+10x-6y-66=0$