Search any question & find its solution
Question:
Answered & Verified by Expert
The line $4 x+3 y-4=0$ divides the circumference of a circle in the ratio $1: 2$. If $\mathrm{C}(5,3)$ is the centre of that circle, then equation of the circle is
Options:
Solution:
1585 Upvotes
Verified Answer
The correct answer is:
$(x-5)^2+(y-3)^2=10^2$
Since line divides the circumference of circle in ratio $1: 2$
$$
\begin{aligned}
& \therefore \angle \mathrm{AOB}=120^{\circ} \\
& \mathrm{OC}=\frac{4(5)+3(3-4)}{\sqrt{16+9}}=\frac{20+9-4}{\sqrt{25}}=5 \\
& \cos 60^{\circ}=\frac{\mathrm{OC}}{\mathrm{OA}}=\frac{1}{2}=\frac{5}{\mathrm{OA}} \Rightarrow \mathrm{OA}=10
\end{aligned}
$$
$\therefore$ Equation of circle is
$$
(x-5)^2+(y-3)^2=(10)^2
$$
$$
\begin{aligned}
& \therefore \angle \mathrm{AOB}=120^{\circ} \\
& \mathrm{OC}=\frac{4(5)+3(3-4)}{\sqrt{16+9}}=\frac{20+9-4}{\sqrt{25}}=5 \\
& \cos 60^{\circ}=\frac{\mathrm{OC}}{\mathrm{OA}}=\frac{1}{2}=\frac{5}{\mathrm{OA}} \Rightarrow \mathrm{OA}=10
\end{aligned}
$$
$\therefore$ Equation of circle is
$$
(x-5)^2+(y-3)^2=(10)^2
$$
Looking for more such questions to practice?
Download the MARKS App - The ultimate prep app for IIT JEE & NEET with chapter-wise PYQs, revision notes, formula sheets, custom tests & much more.