Q186 / 270 · NDA 2021 (Phase 2) · Mathematics
A circle is passing through the points $(5,-8),(-2,9)$ and $(2,1)$.
What are the coordinates of the centre of the circle?
- A$(-2,-50)$
- B$(-50,-20)$
- C$(-24,-58)$
- Correct answerD$(-58,-24)$
Solution
Let equation of circle
$\begin{aligned}
&(x-h)^2+(y-k)^2=r^2 \\
& \therefore(5-h)^2+(-8-k)^2=r^2 \quad ......\mathrm{(i)}\\
&(-2-h)^2+(9-k)^2=r^2 \quad ......\mathrm{(ii)}\\
&(2-h)^2+(1-k)^2=r^2 \quad ......\mathrm{(iii)}
\end{aligned}$
From (i) and (ii),
$\begin{aligned}
& 25+h^2-10 h+64+k^2+16 k \\
& =4+h^2+4 h+81+k^2-18 k-14 h+34 k \\
& =-4 \\
& \Rightarrow-7 h+17 k=-2 \quad ......\mathrm{(iv)}
\end{aligned}$
From (ii) and (iii)
$\begin{aligned}
& 4+h^2+4 h+81+k^2-18 k \\
& =4+h^2-4 h+1+k^2-2 k \\
& =8 h-16 k=-80 \\
& h-2 k=-10 \quad ......\mathrm{(v)}
\end{aligned}$
Solving equations (iv) and (v), we get
$\begin{aligned}
& h=-58 \text { and } k=-24 \\
& \therefore \text { Centre }(-58,-24)
\end{aligned}$
Hence, the correct answer is option: 'D'.