Find the angles between the following pairs of straight lines: 3x… - Vidyadip

Question

Find the angles between the following pairs of straight lines: 3x + 4y - 7 = 0 and 4x - 3y + 5 = 0

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Answer

To find angles between the lines, convert the equations in the form$\text{y}=\text{mx}+\text{c}$
$3\text{x}+4\text{y}-7=0$
$\Rightarrow4\text{y}=-3\text{x}+7$
$\text{y}=\frac{-3}{4}\text{y}+\frac{7}{4}$
$\Rightarrow\text{m}_1=\frac{-3}{4}$
Also, $4\text{x}-3\text{y}+5=0$
$\Rightarrow3\text{y}=4\text{x}+5$
$\Rightarrow\text{y}=\frac{4}{3}\text{x}+\frac{5}{3}$
$\Rightarrow\text{m}_1=\frac{4}{3}$
The angle between the lines is given by
$\tan\theta=\Big|\frac{\text{m}_1-\text{m}_2}{1+\text{m}_1\text{m}_2}\Big|$
$=\Bigg|\frac{\frac{-3}{4}-\frac{4}{3}}{1+\frac{(-3)}{4}\big(\frac{4}{3}\big)}\Bigg|=\Bigg|\frac{\frac{-3}{4}-\frac{4}{3}}{1-1}\Bigg|$
$\Rightarrow\theta=\frac{\pi}{2}$ or $90^\circ$

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