Question: Find the angle between the lines: $2x -3y = 8$ and $3x +2y= -5$


Solution:
Given,

Equation of line 1 = $2x -3y = 8$
Slope of line 1 $(m_1)$ = $ - \dfrac{coefficient of x}{coefficient of y}$
= $ - \dfrac{2}{-3}$
= $\dfrac{2}{3}$

Equation of line 2 = $3x+2y= -5$
Slope of line 2 $(m_2)$ = $ - \dfrac{coefficient of x}{coefficient of y}$
= $ - \dfrac{3}{2}$

Using formula;
Angle between two lines A and B is;
$tan \theta = \left ( \pm \dfrac{m_1 - m_2}{1 + m_1.m_2} \right )$

$or, \theta = tan^{-1} \left ( \pm \dfrac{\dfrac{2}{3} - \left ( - \dfrac{3}{2} \right ) }{1 + \dfrac{2}{3}. \left ( - \dfrac{3}{2} \right )} \right )$

$or, \theta = tan^{-1} \left ( \pm \dfrac{\dfrac{2}{3} - \left ( - \dfrac{3}{2} \right ) }{0} \right ) $

[ Anything divided by 0 is undefined ]

$or, \theta = tan^{-1} \left ( \pm undefined \right )$

Taking positive sign,

$or, \theta = tan^{-1} undefined$

$\therefore, \theta = 90°$

Taking negative sign,

$or, \theta = tan^{-1} (-undefined)$

$\therefore, \theta = 90°$

Therefore, the required angle between the given two lines is either 90° or 90°.


Related Notes and Solutions:

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

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