Question

The angle of rotation of the axes so that the equation $ax+by+c=0$ may be reduced to $X=p$ is

• A. ${\tan }^{-1}\frac{b}{a}$
• B. ${\tan }^{-1}\frac{a}{b}$
• C. $\frac{\pi }{2}$
• D. $\frac{\pi }{3}$

Answer 1

Answer: (A)

${\tan }^{-1}\frac{b}{a}$

Given equation is $ax+by+c=0$ ...(1)

Let the line axes be turned through an angle $\theta$

The equation of transformation are

$x=x^{\prime }\cos \theta -y^{\prime }\sin \theta ,y=x^{\prime }\sin \theta +y^{\prime }\cos \theta$

$\therefore$ equation (1) transformed into

$a(x^{\prime }\cos \theta -y^{\prime }\sin \theta )+b(x^{\prime }\sin \theta +y^{\prime }\cos \theta )+c=0$

$\Rightarrow (a\cos \theta +b\sin \theta )x^{\prime }+(b\cos \theta -a\sin \theta )y^{\prime }+c=0$ ...(2)

The transformed equation will be independent of $y^{\prime }$ if

$b\cos \theta -a\sin \theta =0\Rightarrow \tan \theta =\frac{b}{a}\Rightarrow \theta ={\tan }^{-1}\frac{b}{a}$