Question

Let $L$ be the line $2x+y=2$. If the axes are rotated anti-clockwise by ${45}^o$, then the intercepts made by the line $L$ on the new axes are respectively

• A. $\sqrt{2}$ and $1$
• B. $1$ and $\sqrt{2}$
• C. $2\sqrt{2}$ and $2\sqrt{2}/3$
• D. $2\sqrt{2}/3$ and $2\sqrt{2}$

Answer 1

The correct option is D $2\sqrt{2}/3$ and $2\sqrt{2}$

We have our eq. $2x+y=2$

or

$\frac{x}{2}$ +$\frac{y}{1}$$=$ $1$

By this Form of this equation of line we can clearly see the intercepts of this line on axes i.e $1$ & $2$

now we have to rotate the axis to ${45}^{\circ }$ degrees let's say in clockwise manner

So,

$x=Xcos\left({45}^{\circ }\right)-Ysin\left({45}^{\circ }\right)$

$y=Xsin\left({45}^{\circ }\right)+Ycos\left({45}^{\circ }\right)$

put these values in the equation of line

$\frac{Xcos\left({45}^{\circ }\right)-Ysin\left({45}^{\circ }\right)}{2}$ + $\frac{Xsin\left({45}^{\circ }\right)+Ycos\left({45}^{\circ }\right)}{1}$ $=$ $1$

$\frac{X-Y}{2\sqrt{2}}$+$\frac{X+Y}{\sqrt{2}}$ $=$ $1$

$\frac{3X}{2\sqrt{2}}$ +$\frac{Y}{2\sqrt{2}}$$=$ $1$

Hence Intercepts of this newly formed equation are $\frac{2\sqrt{2}}{3}$ and $2\sqrt{2}$ respectively on the axes.