Find the angle between the lines y = (2 - 3)(x + 5) and y = (2 + … - Vidyadip

Question

Find the angle between the lines $\text{y} = (2 - \sqrt{3})(\text{x} + 5)$ and $\text{y} = (2 + \sqrt{3})(\text{x} - 7).$

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Answer

The given equation are $\text{y}\ =\ (2-\sqrt{3})(\text{x}+5)\ .....(\text{i})$
and $\text{y}=(2+\sqrt{3})(\text{x}-7)$
Slope of $eq. (i)\ m_1\ ($say$)\ =\ (2-\sqrt{3})$
and slope of $eq. (ii)\ m_2\ ($say$)\ =\ (2+\sqrt{3})$
let $\theta$ be the angle between the two given lines
$\therefore \tan\theta=\Big|\frac{\text{m}_1-\text{m}_2}{1+\text{m}_1\text{m}_2}\Big|=\Big|\frac{2-\sqrt{3}-2-\sqrt{3}}{1+(2-\sqrt{3})(2+\sqrt{3})}\Big|$
$=\Big|\frac{-2\sqrt{3}}{1+4-3}\Big|=\Big|\frac{-2\sqrt{3}}{2}\Big|=|-\sqrt{3}|$
$\Rightarrow \tan\theta=\sqrt{3}\text{ or }-\sqrt{3}$
$\therefore \theta = 60^\circ\text{ or }120^\circ$
Hence$,$ the required angle is $60^\circ$ or $120^\circ .$

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