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Find the Equations to the Straight Lines Which Pass Through the Point (H, K) and Are Inclined at Angle Tan−1 M to the Straight Line Y = Mx + C. - Mathematics

Answer in Brief

Find the equations to the straight lines which pass through the point (h, k) and are inclined at angle tan−1 m to the straight line y = mx + c.

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We know that the equations of two lines passing through a point \[\left( x_1 , y_1 \right)\] and making an angle \[\alpha\] with the given line y = m'x + c are \[y - y_1 = \frac{m^{\prime}\pm \tan\alpha}{1 \mp m^{\prime} \tan\alpha}\left( x - x_1 \right)\]


\[x_1 = h, y_1 = k, \alpha = \tan^{- 1} m, m^{\prime}= m\]

So, the equations of the required lines are

\[y - k = \frac{m + m}{1 - m^2}\left( x - h \right) and y - k = \frac{m - m}{1 + m^2}\left( x - h \right)\]

\[ \Rightarrow y - k = \frac{2m}{1 - m^2}\left( x - h \right) \text { and } y - k = 0\]

\[ \Rightarrow \left( y - k \right)\left( 1 - m^2 \right) = 2m\left( x - h \right)\text { and } y = k\]

Concept: Straight Lines - Equation of Family of Lines Passing Through the Point of Intersection of Two Lines
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RD Sharma Class 11 Mathematics Textbook
Chapter 23 The straight lines
Exercise 23.18 | Q 4 | Page 124
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