Question

Find the acute angle between the lines \[ax + by + c = 0\]and \[(a + b)x = (a - b)y\] where\[\left( {a \ne b} \right)\].
$
  {\text{A}}{\text{.3}}{{\text{0}}^{\text{0}}} \\
  {\text{B}}{\text{.6}}{{\text{0}}^{\text{0}}} \\
  {\text{C}}{\text{.1}}{{\text{5}}^{\text{0}}} \\
  {\text{D}}{\text{.4}}{{\text{5}}^{\text{0}}} \\
 $

Answer 1

Hint: When two lines intersect with each other, they form an angle known as the angle between the lines, and if that angle rotates one of the lines then both the lines will coincide. To find the angles between the lines, the first step is to find the slopes of the lines between which the angle to be determined and then find the angle formed by them. For the straight-line equation \[y = mx + c\] where \[m\] is the slope of the line, c is the intercept made on the y-axis. Hence, find the slope of both the equations.

Complete step by step answer:
For the given equations of the line
\[
  ax + by + c = 0 \\
  \left( {a + b} \right)x = (a - b)y \\
 \]
Write the given equation in the form of \[y = mx + c\] ; we get the equations as:
\[
  ax + by + c = 0 \\
  by = - ax - c \\
  y = \dfrac{{ - ax - c}}{b} \\
  y = - \dfrac{a}{b}x - c \\
 \]
And,
\[
  \left( {a + b} \right)x = (a - b)y \\
  y = \dfrac{{\left( {a + b} \right)}}{{\left( {a - b} \right)}}x + 0 \\
 \]
The slope of the line is the coefficient of x when the line equation is written in standard form. Hence, the slope of the lines is:
\[
  {m_1} = - \dfrac{a}{b} \\
  {m_2} = \dfrac{{\left( {a + b} \right)}}{{\left( {a - b} \right)}} \\
 \]
Now, find the angle between the lines by using formula\[\tan \theta = \left| {\dfrac{{{m_1} - {m_2}}}{{1 + {m_1}{m_2}}}} \right|\].
\[
  \tan \theta = \left| {\dfrac{{ - \dfrac{a}{b} - \dfrac{{\left( {a + b} \right)}}{{\left( {a - b)} \right)}}}}{{1 + \left( { - \dfrac{a}{b} \times \dfrac{{\left( {a + b} \right)}}{{\left( {a - b} \right)}}} \right)}}} \right| \\
   = \left| {\dfrac{{\dfrac{{ - {a^2} + ab - ab - {b^2}}}{{ab - {b^2}}}}}{{1 + \dfrac{{\left( { - {a^2} - ab} \right)}}{{\left( {ab - {b^2}} \right)}}}}} \right| \\
   = \left| {\dfrac{{\dfrac{{ - {a^2} - {b^2}}}{{ab - {b^2}}}}}{{\dfrac{{ab - {b^2} - {a^2} - ab}}{{ab - {b^2}}}}}} \right| \\
   = \left| {\dfrac{{\dfrac{{ - {a^2} - {b^2}}}{{ab - {b^2}}}}}{{\dfrac{{ - {a^2} - {b^2}}}{{ab - {b^2}}}}}} \right| \\
   = 1 \\
 \]
\[
  \tan \theta = 1 \\
  \tan \theta = \tan {45^ \circ } \\
  \theta = {45^ \circ } \\
 \], (In trigonometric function \[\tan {45^ \circ } = 1\])

Hence, the angle between the lines is \[{45^0}\]which lies between the ranges \[\left( {0 - {{90}^0}} \right)\]hence the angle between the lines is acute.

Note: To find the angle between the straight lines finds the slope of lines between whose angles is to be found. When angle made between the lines after intersecting are between \[\left( {0 - {{90}^0}} \right)\]then, that angle is known as acute angles whereas if the angle formed is between \[\left( {{{90}^ \circ } - {{180}^ \circ }} \right)\]then the angle is obtuse and if the angle between the lines is ${90^0}$ then, the angle is known as a right-angled triangle.