Question

A line has intercepts $a$ and $b$ on the coordinate axes. When the axes are rotated through an angle $\alpha$, keeping the origin fixed, the line makes equal intercepts on the coordinate axes, then $\tan$$\alpha$=

• A. $\frac{a+b}{a-b}$
• B. $\frac{a-b}{a+b}$
• C. $a^2-b^2$
• D. None of these

Answer 1

The correct option is B $\frac{a-b}{a+b}$

The coordinates of the points where the line cuts the x-axis and y-axis are (a,0) and (0,b) respectively.The equation of the line is $ax+by=ab$.

When the axes are rotated through an angle $\alpha$ the equation becomes

$x(b\cos \alpha +a\sin \alpha )+y(a\cos \alpha -b\sin \alpha )=ab$.

x and y intercepts of the line are $\frac{ab}{b\cos \alpha +a\sin \alpha }$ and $\frac{-ab}{b\sin \alpha -a\cos \alpha }$.

Given that the intercepts are equal.

$\Rightarrow b\cos \alpha +a\sin \alpha =a\cos \alpha -b\sin \alpha$

$\Rightarrow \tan \alpha =\frac{a-b}{a+b}$