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  • Find the angle between the lines y is equal to 2 minus square root 3 x plus 5 and y is equal to 2 plus square root 3 x minus 7

 Find the angle between the lines  y=\left ( 2-\sqrt{3} \right )\left ( x+5 \right )  and y=\left ( 2+\sqrt{3} \right )\left ( x-7 \right )

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Given equations are  y=\left ( 2-\sqrt{3} \right )\left ( x+5 \right )

y=\left ( 2-\sqrt{3} \right )x+\left ( 2-\sqrt{3} \right )5 ................(1)

and   y=\left ( 2+\sqrt{3} \right )\left ( x-7 \right )

y=\left ( 2+\sqrt{3} \right ) x-7 \left ( 2+\sqrt{3} \right )............(2)

 In equation 1 the slope is  \left ( 2-\sqrt{3} \right )as it is in the form of y=mx+b  and in equation 2 it is  2+\sqrt{3} 

 Let θ be the angle between the given m1 and m2  two lines \tan \theta =\left |\frac{\left ( m_{1}-m_{2} \right )}{1+m_{1}m_{2}} \right |

Putting the values of m1 and m2 in above equation  we get  

\tan \theta =\left |\frac{2-\sqrt{3}-\left ( 2+\sqrt{3} \right )}{1+\left ( 2-\sqrt{3} \right )\left (2+\sqrt{3} \right )} \right |

=\left | -\frac{2\sqrt{3}}{1+\left ( 4-3 \right )} \right |

=\left | -\frac{2\sqrt{3}}{2} \right |

= \sqrt{3}

\theta =\tan ^{-1}\sqrt{3}

  θ=600

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