Find the angle between the lines y = (2 - 3)(x + 5) and y = (2 + 3)(x - 7).

The given equation are y=(2-3)(x+5) .....(i) and y=(2+3)(x-7) Slope of eq. (i) m1 (say) =(2-3) and slope of eq. (ii) m2 (say) =(2+3) let be the angle between the two given lines = | m_1-m_2 1+m_1m_2 |= | 2-3-2-3 1+(2-3)(2+3) | = | -23 1+4-3 |= | -23 2 |=|-3| =3 or -3 = 60^ or 120^ Hence, the required angle is 60° or 120°.

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

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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) m₁ (say) $=(2-\sqrt{3})$
and slope of eq. (ii) m₂ (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° or 120°.

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