Question

The combined equation to a pair of straight lines passing through the origin and inclined at an
angles 30° and 60° respectively with X-axis is

​

Answer 1

$\textbf{Given:}$

$\text{Angle of inclinations are 30^{\circ} and 60^{\circ} }$

$\textbf{To find:}$

$\text{Combined equation of pair of st.lines passing through origin}$

$\textbf{Solution:}$

$\textbf{For Line1:}$

$\text{Angle of inclination}\;\theta_1=30^{\circ}$

$\text{Slope}\;m_1=\tan{\theta_1}=\tan{30^{\circ}}=\dfrac{1}{\sqrt{3}}$

$\text{Equation Line1 is}$

$y=m_1\,x$

$y=\dfrac{1}{\sqrt{3}}\,x$

$\sqrt{3}y=x$

$\implies\bf\,x-\sqrt{3}y=0$

$\textbf{For Line2:}$

$\text{Angle of inclination}\;\theta_2=60^{\circ}$

$\text{Slope}\;m_2=\tan{\theta_2}=\tan{60^{\circ}}=\sqrt{3}$

$\text{Equation Line2 is}$

$y=m_2\,x$

$y=\sqrt{3}\,x$

$\implies\bf\,\sqrt{3}\,x-y=0$

$\textbf{Combined equation is}$

$(x-\sqrt{3}y)(\sqrt{3}\,x-y)=0$

$\sqrt{3}\,x^2-xy-3xy+\sqrt{3}\,y^2=0$

$\implies\boxed{\bf\sqrt{3}\,x^2-4xy+\sqrt{3}\,y^2=0}$

Find more:

Find p and q, if the following equation represents a pair of perpendicular lines

2x²+ 4xy – py² + 4x +qy+1=0​

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