Show that the equations to the straight lines passing through the point $(3,-2)$ and inclined at ${60}^{\circ}$ to the line $\sqrt{3}x+y=1$ are $y+2=0$ and $y-\sqrt{3}x+2+3\sqrt{3}=0$

kennadiceKesezt
2022-09-25
Answered

Show that the equations to the straight lines passing through the point $(3,-2)$ and inclined at ${60}^{\circ}$ to the line $\sqrt{3}x+y=1$ are $y+2=0$ and $y-\sqrt{3}x+2+3\sqrt{3}=0$

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vidovitogv5

Answered 2022-09-26
Author has **10** answers

Let $\theta $ be the angle between two lines whose slopes are ${m}_{1},{m}_{2}$. Then we have:

$\mathrm{tan}\theta =|\frac{{m}_{1}-{m}_{2}}{1+{m}_{1}{m}_{2}}|$

$\mathrm{tan}\theta =|\frac{{m}_{1}-{m}_{2}}{1+{m}_{1}{m}_{2}}|$

As we know ${m}_{1}$, we can just plug in the value and get ${m}_{2}$:

$\sqrt{3}=|\frac{-\sqrt{3}-{m}_{2}}{1-\sqrt{3}{m}_{2}}|$

$\sqrt{3}-3{m}_{2}=-\sqrt{3}-{m}_{2}\phantom{\rule{thickmathspace}{0ex}}\u27f9\phantom{\rule{thickmathspace}{0ex}}{m}_{2}=\sqrt{3}$

or

$-\sqrt{3}+3{m}_{2}=-\sqrt{3}-{m}_{2}\phantom{\rule{thickmathspace}{0ex}}\u27f9\phantom{\rule{thickmathspace}{0ex}}{m}_{2}=0$

$y+2=0(x-3)\phantom{\rule{thickmathspace}{0ex}}\u27f9\phantom{\rule{thickmathspace}{0ex}}y+2=0$

$y+2=\sqrt{3}(x-3)\phantom{\rule{thickmathspace}{0ex}}\u27f9\phantom{\rule{thickmathspace}{0ex}}y-\sqrt{3}x+2+3\sqrt{3}=0$

$\mathrm{tan}\theta =|\frac{{m}_{1}-{m}_{2}}{1+{m}_{1}{m}_{2}}|$

$\mathrm{tan}\theta =|\frac{{m}_{1}-{m}_{2}}{1+{m}_{1}{m}_{2}}|$

As we know ${m}_{1}$, we can just plug in the value and get ${m}_{2}$:

$\sqrt{3}=|\frac{-\sqrt{3}-{m}_{2}}{1-\sqrt{3}{m}_{2}}|$

$\sqrt{3}-3{m}_{2}=-\sqrt{3}-{m}_{2}\phantom{\rule{thickmathspace}{0ex}}\u27f9\phantom{\rule{thickmathspace}{0ex}}{m}_{2}=\sqrt{3}$

or

$-\sqrt{3}+3{m}_{2}=-\sqrt{3}-{m}_{2}\phantom{\rule{thickmathspace}{0ex}}\u27f9\phantom{\rule{thickmathspace}{0ex}}{m}_{2}=0$

$y+2=0(x-3)\phantom{\rule{thickmathspace}{0ex}}\u27f9\phantom{\rule{thickmathspace}{0ex}}y+2=0$

$y+2=\sqrt{3}(x-3)\phantom{\rule{thickmathspace}{0ex}}\u27f9\phantom{\rule{thickmathspace}{0ex}}y-\sqrt{3}x+2+3\sqrt{3}=0$

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