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Case Study - 2
In a society, there is a circular park having two gates. The gates are placed at points $A(10, 20)$ and $B(50, 50)$, as shown in the figure below. Two fountains are installed at points $P$ and $Q$ on $AB$ such that $AP = PQ = QB$.
Based on the above information, answer the following questions :
(i) Find the coordinates of the centre $C$.
(ii) Find the radius of the circular park.
(iii) (a) Find the coordinates of the point $P$.
OR
(b) Find the distance of the fountain at $Q$ from gate $A$.
In a society, there is a circular park having two gates. The gates are placed at points $A(10, 20)$ and $B(50, 50)$, as shown in the figure below. Two fountains are installed at points $P$ and $Q$ on $AB$ such that $AP = PQ = QB$.
Based on the above information, answer the following questions :
(i) Find the coordinates of the centre $C$.
(ii) Find the radius of the circular park.
(iii) (a) Find the coordinates of the point $P$.
OR
(b) Find the distance of the fountain at $Q$ from gate $A$.
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Solution: (i) Co-ordinates of $C$ are $(\frac{10 + 50}{2}, \frac{20 + 50}{2}) = C(30, 35)$
(ii) Radius $= \sqrt{(30 - 10)^2 + (35 - 20)^2} = 25$
(iii) (a) $P$ divides $AB$ in the ratio $1 : 2$, co-ordinates of $P$ are $(\frac{1 \times 50 + 2 \times 10}{3}, \frac{1 \times 50 + 2 \times 20}{3})$ i.e. $(\frac{70}{3}, 30)$
OR
(b) Distance $AB = 2 \times 25 = 50$
$AQ = \frac{2}{3}AB = \frac{2}{3} \times 50$
$AQ = \frac{100}{3}$
(ii) Radius $= \sqrt{(30 - 10)^2 + (35 - 20)^2} = 25$
(iii) (a) $P$ divides $AB$ in the ratio $1 : 2$, co-ordinates of $P$ are $(\frac{1 \times 50 + 2 \times 10}{3}, \frac{1 \times 50 + 2 \times 20}{3})$ i.e. $(\frac{70}{3}, 30)$
OR
(b) Distance $AB = 2 \times 25 = 50$
$AQ = \frac{2}{3}AB = \frac{2}{3} \times 50$
$AQ = \frac{100}{3}$