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# Find the equation of the line which is at a distance of 3 units from the origin

Question:

Find the equation of the line which is at a distance of 3 units from the origin such that tan $\propto=\frac{5}{12}$

here ∝ is the acute angle which this perpendicular makes with the positive direction of the x-axis.

Solution:

To Find:The equation of the line.

Given : $\propto=\frac{5}{12}$ and $p=3$ units.

Since $\tan \alpha=\frac{\text { opp }}{\text { adj }}=\frac{5}{12}$

Using Pythagoras theorem :

hyp $=\sqrt{25+144}=\sqrt{169}=13$ units.

From the figure: $\cos \propto=\frac{\text { adj }}{\text { hyp }}=\frac{12}{13}$ and $\sin \propto=\frac{\text { opp }}{\text { hyp }}=\frac{5}{13}$

Formula used:

equation of the line: $x \cos \alpha+y \sin \propto=p$

$x \times\left(\frac{12}{13}\right)+y \times\left(\frac{5}{13}\right)=5$

Hence, $12 x+5 y=65$ is the required equation of the line.