# Find the Point of Intersection of the Following Pairs of Lines: Y = M 1 X + a M 1 and Y = M 2 X + a M 2 . - Mathematics

Find the point of intersection of the following pairs of lines:

$y = m_1 x + \frac{a}{m_1} \text { and }y = m_2 x + \frac{a}{m_2} .$

#### Solution

The equations of the lines are $y = m_1 x + \frac{a}{m_1} \text { and } y = m_2 x + \frac{a}{m_2} .$

Thus, we have:

$m_1 x - y + \frac{a}{m_1} = 0$               ... (1)

$m_2 x - y + \frac{a}{m_2} = 0$              ... (2)

Solving (1) and (2) using cross-multiplication method:

$\frac{x}{- \frac{a}{m_2} + \frac{a}{m_1}} = \frac{y}{\frac{a m_2}{m_1} - \frac{a m_1}{m_2}} = \frac{1}{- m_1 + m_2}$

$\Rightarrow x = \frac{\frac{- a}{m_2} + \frac{a}{m_1}}{- m_1 + m_2}, y = \frac{\frac{a m_2}{m_1} - \frac{a m_1}{m_2}}{- m_1 + m_2}$

$\Rightarrow x = \frac{a}{m_1 m_2} \text { and }y = \frac{a\left( m_1 + m_2 \right)}{m_1 m_2}$

Hence, the point of intersection is $\left( \frac{a}{m_1 m_2}, \frac{a\left( m_1 + m_2 \right)}{m_1 m_2} \right) \text { or }\left( \frac{a}{m_1 m_2}, a\left( \frac{1}{m^1} + \frac{1}{m_2} \right) \right)$.

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#### APPEARS IN

RD Sharma Class 11 Mathematics Textbook
Chapter 23 The straight lines
Exercise 23.1 | Q 1.3 | Page 77