118
Case Study - 3
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.
Show SolutionHide Solution↓
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 $AB = 2 \times 25 = 50$
$AQ = \frac{2}{3} AB = \frac{2}{3} \times 50$
$AQ = \frac{100}{3}$