First slide

Cartesian plane

Question

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

Moderate

A

$\frac{a + b}{a - b}$
B

$\frac{a - b}{a + b}$
C

$a^{2} - b^{2}$
D

none of these

Solution

Let the equation of the line in the original
coordinate system be $\frac{x}{a} + \frac{y}{b} = 1$ . If (X,Y)
denote the coordinates
of any point P (x, y) in the new coordinate system
obtained by rotation of the axes through an angle a, then
$x = X \cos α - Y \sin α, y = X \sin α + Y \cos α$
So that the equation of the line with reference to new system of coordinates is
$\frac{X \cos α - Y \sin α}{a} + \frac{X \sin α + Y \cos α}{b} = 1$
or $X (\frac{\cos α}{a} + \frac{\sin α}{b}) + Y (\frac{\cos α}{b} - \frac{\sin α}{a}) = 1$
Since it makes equal intercepts on the coordinates axes.
$b \cos α + a \sin α = a \cos α - b \sin α$
$\Rightarrow (a - b) \cos α = (a + b) \sin α$
$\Rightarrow \tan α = \frac{a - b}{a + b}$

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