If a line passes through the midpoint of $A B$ where $A$ is $(3, 0)$ and $B$ is $(5, 4)$ and makes an angle of $60^{\circ}$ with $X -$ axis.Find its equation.

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Solution

Midpoint $= (\frac{x_{1} + x_{2}}{2}, \frac{y_{1} + y_{2}}{2})$
Midpoint of $A B = (\frac{3 + 5}{2}, \frac{0 + 4}{2}) = (4, 2)$
$θ = 60^{\circ}$
Slope $= m = tan 60^{\circ} = \sqrt{3}$
Equation of a line is $y - y_{1} = m (x - x_{1})$
$y - 2 = \sqrt{3} (x - 4)$
$y - 2 = \sqrt{3} x - 4 \sqrt{3}$
Equation of a line is $\sqrt{3} x - y + 2 - 4 \sqrt{3} = 0$

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If a line passes through the midpoint of AB where A is (3,0) and B is (5,4) and makes an angle of 60∘ with X−axis.Find its equation.

Midpoint=(x1+x22,y1+y22)Midpoint of AB=(3+52,0+42)=(4,2)θ=60∘Slope=m=tan60∘=√3Equation of a line is y−y1=m(x−x1)y−2=√3(x−4)y−2=√3x−4√3Equation of a line is √3x−y+2−4√3=0

If a line passes through the midpoint of $A B$ where $A$ is $(3, 0)$ and $B$ is $(5, 4)$ and makes an angle of $60^{\circ}$ with $X -$ axis.Find its equation.