Two lines passing through the point $(2, 3)$ intersect each other at an angle of $60^\circ$. If the slope of one line is $2$. Find the equation of other line.

Question

Accepted Answer

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> [!TLDR]
> $	an60^\circ=\sqrt3=\left|\frac{m-2}{1+2m}ight|$ gives two slopes; each with point $(2,3)$ gives $y-3=m(x-2)$.
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> Let the other line have slope $m$. The angle $	heta$ between two lines of slopes $m$ and $2$ satisfies
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> $	an	heta = \left|\dfrac{m - 2}{1 + 2m}ight|$.
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> With $	heta = 60^\circ$, $	an60^\circ = \sqrt3$:
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> $\left|\dfrac{m-2}{1+2m}ight| = \sqrt3$.
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> **Case 1:** $\dfrac{m-2}{1+2m} = \sqrt3 \Rightarrow m - 2 = \sqrt3(1+2m) \Rightarrow m - 2\sqrt3 m = 2 + \sqrt3 \Rightarrow m = \dfrac{2+\sqrt3}{1-2\sqrt3}$.
> …

Slope value	What the line does
$m > 0$	Rises left to right (uphill)
$m < 0$	Falls left to right (downhill)
$m = 0$	Horizontal (flat)
$m$ undefined	Vertical (straight up/down)

Q.Two lines passing through the point $(2, 3)$ intersect each other at an angle of $60^\circ$ . If the slope of one line is $2$ . Find the equation of other line.

Concept understanding — Slope Calculation

Slope Calculation — From Intuition to Precision

The Intuition: Rise Over Run

The Precise Definition

What the Number Tells You

A Worked Example

Q.Two lines passing through the point (2,3)(2, 3)(2,3) intersect each other at an angle of 60∘60^\circ60∘. If the slope of one line is 222. Find the equation of other line.

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Q.Two lines passing through the point $(2, 3)$ intersect each other at an angle of $60^\circ$ . If the slope of one line is $2$ . Find the equation of other line.

Unlock everything free for 14 days