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Case Study - 3
38. There is a semicircular park in Aman's society. He wishes to plant saplings along the boundary of the park. There is a borewell at the centre $O$ of the park along the diameter $AB$ as shown in the figure below. Based on the above information, answer the following questions : (i) Find the coordinates of point $O$. (ii) Find the radius of the semicircular park. (iii) (a) One sapling is kept at point $C(12, y)$. Find the coordinates of $C$. OR (b) One sapling is kept at point $P$ along $AB$ so that $PA = \frac{1}{3} PB$. Find the coordinates of $P$.
38. There is a semicircular park in Aman's society. He wishes to plant saplings along the boundary of the park. There is a borewell at the centre $O$ of the park along the diameter $AB$ as shown in the figure below. Based on the above information, answer the following questions : (i) Find the coordinates of point $O$. (ii) Find the radius of the semicircular park. (iii) (a) One sapling is kept at point $C(12, y)$. Find the coordinates of $C$. OR (b) One sapling is kept at point $P$ along $AB$ so that $PA = \frac{1}{3} PB$. Find the coordinates of $P$.
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(i) Coordinates of $O$ are $(12, 3)$ (1 mark)
(ii) Radius $= 10$ (1 mark)
(iii) (a) $OC = 10 \Rightarrow y = 13$ or $y = -7$. $C$ is $(12, 13)$ or $(12, -7)$ ($\frac{1}{2} + 1 + \frac{1}{2}$ marks) OR (b) $P$ divides $AB$ in ratio $1:3$. $P = \left( \frac{1 \times 22 + 3 \times 2}{4}, \frac{1 \times 3 + 3 \times 3}{4} \right) = (7, 3)$ ($\frac{1}{2} + 1 + \frac{1}{2}$ marks)
(ii) Radius $= 10$ (1 mark)
(iii) (a) $OC = 10 \Rightarrow y = 13$ or $y = -7$. $C$ is $(12, 13)$ or $(12, -7)$ ($\frac{1}{2} + 1 + \frac{1}{2}$ marks) OR (b) $P$ divides $AB$ in ratio $1:3$. $P = \left( \frac{1 \times 22 + 3 \times 2}{4}, \frac{1 \times 3 + 3 \times 3}{4} \right) = (7, 3)$ ($\frac{1}{2} + 1 + \frac{1}{2}$ marks)