Question

The co-ordinates of two points $P$ and $Q$ are $(x_1,y_1)$ and $(x_2,y_2)$ and $O$ is the origin. If circles be described on $OP$ and $OQ$ as diameters then length of their common chord is

• A. $\frac{|x_1{y}_2+x_2{y}_1|}{PQ}$
• B. $\frac{|x_1{y}_2-x_2{y}_1|}{PQ}$
• C. $\frac{|x_1{x}_2+y_1{y}_2|}{PQ}$
• D. $\frac{|x_1{x}_2+y_1{y}_2|}{PQ}$

Answer 1

Answer: (B)

$\frac{|x_1{y}_2-x_2{y}_1|}{PQ}$

Area of triangle with coordinates $P(x_1,y_1),Q(x_2,y_2)\text{and}O(x_3,y_3)$ is

$\Delta POQ=\frac{1}{2}|x_1(y_2-y_3)+x_2(y_3-y_1)+x_3(y_1-y_2)|$

Now,

$OR=\frac{2areaof\Delta OPQ}{PQ}$

$=\frac{2\cdot \left|\frac{1}{2}(x_1{y}_2-x_2{y}_1)\right|}{PQ}$

$=\frac{|x_1{y}_2-x_2{y}_1|}{PQ}$