The line $2x-y=5$ turns about a point....

Question

The line $2x-y=5$ turns about a point on it, whose ordinate and abscissae are equal, through an angle of $45°$, in anti clockwise direction. Find the equation of line in the new position.

My attempt to solve: Let AB be the line with the equation $2x-y=5$ and $B(a,a)$ be a point on it whose ordinate and abscissae are equal. Then $$2x-y=5$$ $$2a-a=5$$ $$a=5$$ Hence, $B(a,a)=B(5,5)$.

My solution ends up here. What should I do next?

Hint: use the formula for angle between lines in order to find the slope of second line - $\displaystyle \tan(\alpha)=\left|\frac{m_1-m_2}{1+m_1m_2}\right|$. Using the fact it is anti clockwise, you can choose the correct sign. — Galc127, Feb 10 '16 at 16:40

score 2 · Answer 1 · edited Feb 10 '16 at 18:40

2

hint use rotation matrix which goes as $$\left(\begin{matrix} \cos45&\sin45\\ -\sin45 & \cos 45\end {matrix} \right)$$ and represent line as a form of matrix multiply these two matrices and get the equation .

edited Feb 10 '16 at 18:40

Harish Chandra Rajpoot

37,450

answered Feb 10 '16 at 16:37

Archis Welankar

15,910

1

In addition to the answer, the line could be represented in parametric form , set x=t, then y= ...t... and perform the matrix multiplication... – imranfat Feb 10 '16 at 16:40
1

ofcourse why not we can also use$ x=rcos\theta,y=rsin\theta$ – Archis Welankar Feb 10 '16 at 16:43
1

Yes, true. I didn't know if OP wanted to go into trigo... – imranfat Feb 10 '16 at 16:48
#Imranfat and Archis, That would be good but I am not familiar with solving it using trigonometry. – Iaamuser user Feb 11 '16 at 02:29

Harish Chandra Rajpoot · Accepted Answer · 2016-02-10T16:43:26.947

1

In new position the line will pass through the point of rotation $(5, 5)$ & the slope of the new line becomes $m=\tan\left(\tan^{-1}(2)+\frac{\pi}{4}\right)=-3$ hence the new equation of the line, using point-slope formula: $y-y_1=m(x-x_1)$,

$$y-5=\left(-3\right)(x-5)$$ $$3x+y=20$$

edited Feb 10 '16 at 16:43

answered Feb 10 '16 at 16:41

Harish Chandra Rajpoot

37,450

score 1 · Answer 3 · answered Feb 10 '16 at 16:41

The line turns by an angle of $45°$ anti-clockwise about the point $(5,5)$. Thus, you need to find the equation of a line whose angle with the $x-axis$ is $45°$ more than this line and which passes through the point $(5,5)$.

The line $2x-y=5$ has a slope of $2$, i.e., it makes an angle of $\arctan2$ with the $x-axis$.

Slope of the required line=$\tan(\frac\pi4+\arctan2)=\frac{1+2}{1-2}=-3$

Its equation, therefore, is $$y-5=-3(x-5)$$ or $$3x+y=20$$

The line $2x-y=5$ turns about a point....

3 Answers3