Answer to Question #146837 in Analytic Geometry for Dhruv rawat

Question

Find the new equation of straight line 2x+y=5 after rotating the axes through 45°.

Answer 1

We have to rotate the axes to $45^0$. Let say in anti-clockwise manner.

So,

$x=x\cos \theta-y \sin \theta$ , $y=x \sin \theta+y \cos \theta$ .

$\implies x=x \cos 45^0-y \sin 45^0=\frac{\sqrt2}{2}x - \frac{\sqrt2}{2}y$

Also,

$y=x\sin 45^0 + y \cos 45^0= \frac{\sqrt2}{2}x + \frac{\sqrt2}{2}y$

Put the new values of $x$ and $y$ into the straight line equation.

$\implies 2(\frac{\sqrt2}{2}x - \frac{\sqrt2}{2}y)+(\frac{\sqrt2}{2}x + \frac{\sqrt2}{2}y)=5\\ \implies x\sqrt2-y\sqrt2+\frac{\sqrt2}{2}x + \frac{\sqrt2}{2}y=5\\ \implies \frac{3\sqrt2}{2}x - \frac{\sqrt2}{2}y=5$

The new equation of the line is $\frac{3\sqrt2}{2}x - \frac{\sqrt2}{2}y=5.$