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Mathematics · 2026 · 3 marks
CBSE 2026 · Region 5 · Set 1 · Q29
(a) A circle centered at (2,1) passes through the points A(5, 6) and B(-3, K). Find the value(s) of K. Hence find length of chord AB.
(a)
A circle centered at $\displaystyle (2,1)$ passes through the points A$\displaystyle (5, 6)$ and B(-$\displaystyle 3$, K). Find the value(s) of K. Hence find length of chord AB.
OR
(b)
Prove that the point P dividing the line segment joining the points $\displaystyle \mathrm{A}(-1,7)$ and $\displaystyle \mathrm{B}(4,-3)$ in the ratio $\displaystyle 3: 2$, lies on the line $\displaystyle x-3 \mathrm{y}=-1$. Also find length of PA and PB.
Marking-scheme solution
Let centre be $\displaystyle \mathrm{O}(2,1) \Rightarrow \mathrm{OA}=\mathrm{OB}$
\[\begin{array}{l}
\sqrt{(5-2)^{2}+(6-1)^{2}}=\sqrt{(-3-2)^{2}+(\mathrm{K}-1)^{2}} \\
\Rightarrow 9=(\mathrm{K}-1)^{2} \\
\Rightarrow \mathrm{~K}=-2,4
\end{array}
\]
For $\displaystyle \mathrm{K}=-2, \mathrm{AB}=\sqrt{128}$ or $\displaystyle 8 \sqrt{2}$
For $\displaystyle \mathrm{K}=4, \mathrm{AB}=\sqrt{68}$ or $\displaystyle 2 \sqrt{17}$
Prove that the point P dividing the line segment joining the points $\displaystyle \mathrm{A}(-1,7)$ and $\displaystyle \mathrm{B}(4,-3)$ in the ratio $\displaystyle 3: 2$, lies on the line $\displaystyle x-3 \mathrm{y}=-1$. Also find length of PA and PB.
X_Mathematics_041_30/$\displaystyle 5$/1_2025-$\displaystyle 26$
$\displaystyle \mathrm{AP}: \mathrm{PB}=3: 2$
Coordinates of $\displaystyle \mathrm{P}=\left(\frac{12-2}{5}, \frac{-9+14}{5}\right)=(2,1)$
Substituting $\displaystyle \mathrm{x}=2$ and $\displaystyle \mathrm{y}=1$ in the given equation
L.H.S. $\displaystyle =x-3 y$
\[\begin{array}{l}
=2-3(1) \\
=-1=\mathrm{R} . \mathrm{H} . \mathrm{S} .
\end{array}
\]
$\displaystyle \therefore \mathrm{P}$ lies on the given line
\[\begin{array}{l}
\mathrm{PA}=\sqrt{(2+1)^{2}+(1-7)^{2}}=\sqrt{45} \text { or } 3 \sqrt{5} \\
\mathrm{~PB}=\sqrt{(2-4)^{2}+(1+3)^{2}}=\sqrt{20} \text { or } 2 \sqrt{5}
\end{array}
\]
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