Question

A straight line has its extremities on two fixed straight lines and cuts off from them a triangle of constant area $C^2$ . Then the locus of the middle point of the line is

• A. $2xy=C^2$
• B. $xy+C^2=0$
• C. $4x^2{y}^2=C_2$
• D. None of these

Answer 1

Answer: (A)

Let A and B are the extremities on two fixed straight line of the given line

$\therefore$ Coordinates of $A(a,0)$ and $B(0,b)$
Let $(h,k)$ is the mid-point of $AB$
$\therefore h=\frac{a}{2}$ and $k=\frac{b}{2}$
Now, area of $\Delta AOB=\frac{1}{2}ab=c^2$
On putting the value of $a$ and $b$, we get
$\frac{1}{2}(2h)(2k)=c^2$
$\Rightarrow 2hk=c^2$
$\therefore$ Locus of the mid-point of the line is $2xy=c^2$