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^2y^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 $2xy=c^{2}$