Question

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

• A.
• B.
• C.
• D.

Answer 1

The correct option is A

Let the given straight line be axis of the co-ordinates and let the equation of the variable line is $\frac{x}{a}+\frac{y}{b}=1$

This line cuts the co-ordinates axis at the point A$(a,0)$ and B$(0,b)$

Therefore the area of the triangle $\equiv \frac{1}{2}ab\Rightarrow$constant

$\frac{1}{2}ab=2c^2$

$\Rightarrow ab=4c^2$ - - - - - - - (1)

If (h,k) be the co-ordinates of the middle point of AB,then

h=$\frac{0+a}{2},k=\frac{0+b}{2}$

h=$\frac{a}{2},k=\frac{b}{2}$

On eliminating a & b from equation(1)

we get,

$a,b=4c^2$

$2h\times 2k=4c^2$

$hk=c^2$

Hence the locus of (h,k) is $xy=c^2$