2016-I-10 – Solving Master

Ans: (a)

4 x - 3 y - 24 = 0

(b) Yes

Let $P = (x, y)$ .
$\begin{array}{rcl} P A & = & P B \\ \sqrt{(5 - x)^{2} + (7 - y)^{2}} & = & \sqrt{(13 - x)^{2} + (1 - y)^{2}} \\ 25 - 10 x + x^{2} + 49 - 14 y + y^{2} & = & 169 - 26 x + x^{2} + 1 - 2 y + y^{2} \\ 16 x - 12 y - 96 & = & 0 \\ 4 x - 3 y - 24 & = & 0 \end{array}$

Therefore, the equation of $Γ$ is $4 x - 3 y - 24 = 0$ .
Note that $H = (6, 0)$ and $K = (0, 8)$ . Note also that $∠ H O K = 90^{\circ}$ . Therefore by the converse of angle in semi circle, $H K$ is a diameter of the circle $C$ . Hence, the radius
$\begin{array}{cl} = & \frac{1}{2} \times \sqrt{(6 - 0)^{2} + (0 - 8)^{2}} \\ = & 5 \end{array}$

Therefore, the circumference of $C$

$\begin{array}{cl} = & 2 π (5) \\ = & 10 π \\ = & 31.4 \\ > & 30 \end{array}$

Hence, the claim is correct.