Question: Find the value of k so that kx + 5y = 10 makes an angle of 30° with y-axis.

Solution:
Given,

Equation of line is $kx + 5y = 10$

Slope of the line ($m_1$) = $- \dfrac{coefficient\; of \;x}{coefficient \; of \; y}$

$= - \dfrac{k}{5}$

Also,
The line makes an angle of 30° with y-axis. If the angle made is positive then $\theta = 30°$ but if the angle made is negative then $\theta = (180°-30°)$

So,

Slope of line ($m_2$) = $tan 30°$
$ = \dfrac{1}{√3}$

Slope of line ($m_3$) = $tan (180°-30°)$
$= - tan 30°$
$= - \dfrac{1}{√3}$


Since, ($m_1\; and \;m_2$) and ($m_2 \;and\; m_3$) represent the slope of the same line, they are equal.

Now,

$m_1 = m_2$
$or, - \dfrac{k}{5} = \dfrac{1}{√3}$

$or, \dfrac{k}{5} = - \dfrac{1}{√3}$

$\therefore k = - \dfrac{5}{√3}$


Also,

$m_1 = m_3$
$or, -\dfrac{k}{5} = -\dfrac{1}{√3}$

$or, \dfrac{k}{5} = \dfrac{1}{√3}$

$\therefore k = \dfrac{5}{√3}$


Hence, the possible values of k are $\pm \dfrac{5}{\sqrt{3}}$


Related Notes and Solutions:

Here is the website link to all the important formulae of Coordinate Geometry of Class 10.

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