Question: The angle between two lines is 45°. If the slope of one of them is \dfrac{1}{2}, find the slope of the other.
Solution:
Given,
Angle between two lines (\theta) = 45°
Slope of one (m_1) = \dfrac{1}{2}
To find: slope of another (m_2) = ?
We know,
tan \theta = \left ( \pm \dfrac{m_1 - m_2}{1 + m_1×m_2} \right )
or, tan 45° = \left ( \pm \dfrac{m_1 - m_2}{1 + m_1×m_2} \right )
or, 1 = \left ( \pm \dfrac{m_1 - m_2}{1 + m_1×m_2} \right )
or, 1 + m_1 × m_2 = \pm ( m_1 - m_2)
or, 1 + \dfrac{1}{2} × m_2 = \pm left ( \dfrac{1}{2} - m_2 \right )
Taking positive sign;
or, 1 + \dfrac{1}{2} × m_2 = \dfrac{1}{2} - m_2
or, 1 + \dfrac{m_2}{2} = \dfrac{1 - 2m_2}{2}
or, \dfrac{2 + m_2}{2} = \dfrac{1 - 2m_2}{2}
or, 2 + m_2 = 1 - 2m_2
or, 2m_2 + m_2 = 1-2
or, 3m_2 = -1
\therefore m_2 = - \dfrac{1}{3}
Taking negative sign,
or, 1 + \dfrac{1}{2} × m_2 = - \left ( \dfrac{1}{2} - m_2 \right )
or, \dfrac{2 + m_2}{2} = - \left ( \dfrac{1 - 2m_2}{2} \right )
or, 2 + m_2 = - ( 1 - 2m_2)
or, 2 + m_2 = -1 + 2 m_2
or, 2m_2 - m_2 = 2 + 1
\therefore m_2 = 3
Hence, the slope of the other line is either 3 or - \dfrac{1}{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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