Question

Find the equation of the straight line at a distance of 3 units from the origin such that the perpendicular from the origin to the line angle a given by the equation are a $tan^{−1} \dfrac{5}{12}$ with the positive direction of the axis of $x$.

Answer 1

The equation of straight line, using trigonometric form can be given as $xcos⁡z+ysin⁡z=p$

where $p$ is the distance of the line from origin. And as the trigonometric value is also given from that we can calculate various other trigonometric ratios and hence put it in the equation of straight line and hence the required solution will be obtained.

Complete step-by-step answer:

As the given information is as straight line at a distance of 3 units from the origin and the trigonometric ratio is given as

$z=tan^{-1} \dfrac{5}{12}$

Diagram:

Taking inverse of the function,

$\Rightarrow \tan z=\dfrac{5}{12}$

Calculating the value of $\cos z$ and $\sin z$ using above ratio and so,

As we know that $\tan z=\dfrac{\sin z}{\cos z}=\dfrac{x}{y}$

And hence, the value can be given as

$\sin z=\dfrac{x}{\sqrt{x^{2}+y^{2}}}$ and similarly the value of $\cos z=\dfrac{y}{\sqrt{x^{2}+y^{2}}}$

Using the above concept and calculating the value as shown below,

$\Rightarrow \sin z=\dfrac{5}{\sqrt{5^{2}+12^{2}}}$

On expanding the term,

$\Rightarrow \sin z=\dfrac{5}{\sqrt{25+144}}=\dfrac{5}{\sqrt{169}}$

Taking square root of the denominator as,

$\Rightarrow \sin z=\dfrac{5}{13}$

Similarly, for $\cos z$

$\Rightarrow \cos z=\dfrac{12}{\sqrt{5^{2}+12^{2}}}$

Hence, on expanding the term

$\Rightarrow \cos z=\dfrac{12}{\sqrt{25+144}}=\dfrac{12}{\sqrt{169}}$

Now taking square root of above term,

$\Rightarrow \cos z=\dfrac{12}{13}$

And hence the distance for the origin is $3 .$

Now, putting all the values in the equation of straight of $x \cos z+y \sin z=p$

Hence, on substituting, we get,

$\Rightarrow x \dfrac{12}{13}+y \dfrac{5}{13}=3$

Hence on simplifying, we get,

$\Rightarrow 12 x+5 y=3(13)$

On expanding, we get,

$\Rightarrow 12 x+5 y=39$

Hence, the equation of line is $12 x+5 y=39$.

Note: In geometry, the notion of line or straight line was introduced by ancient mathematicians to represent straight objects with negligible width and depth. Lines are an idealization of such objects, which are often described in terms of two points or referred to using a single letter.

Using the form to write the equation of line $xcosz+ysinz=p$ to write the equation and hence use the trigonometric ratios also the required answer can be obtained. Calculate the respected values without any mistakes and put in the equation of given lines and so the required equation of line can be calculated