Question

Find the value of \[\lambda \] such that the straight line \[\left( 2x+3y+4 \right)+\lambda (6x-y+12)=0\] is parallel to the y-axis.

Answer 1

Hint: The slope of the line which is parallel to y axis will be \[\dfrac{1}{0}\]. Simplify the given equation and find slope from the equation using formula \[slope=\dfrac{-coefficient\ of\ y}{coefficient\ of\ x}\] and equate it to \[\dfrac{1}{0}\] to get the value of \[\lambda \].

Complete step-by-step answer:
As mentioned in the question, we have to find the value of \[\lambda \].
Now, the given equation can be modified as follows
\[\begin{align}
  & \left( 2x+3y+4 \right)+\lambda 6x-\lambda y+\lambda 12=0 \\
 & x(2+\lambda 6)+y(3-\lambda )+4+\lambda 12=0 \\
\end{align}\]
Now, using the formula for calculating the slope of the given line as follows
\[slope=\dfrac{\lambda -3}{2+6\lambda }\]
Now, this slope should be equal to \[\dfrac{1}{0}\] .
\[\begin{align}
  & \dfrac{\lambda -3}{2+6\lambda }=\dfrac{1}{0} \\
 & 2+6\lambda =0 \\
 & \lambda =\dfrac{-1}{3} \\
\end{align}\]
Hence, this is the value of \[\lambda \] is =\[\dfrac{-1}{3}\].

Note: The students can make an error if they don’t know the formula to calculate the slope of a line which is mentioned in the hint as follows- A line is simply an object in geometry that is characterized under zero width objects that extends on both sides. A straight line is just a line with no curves. So, a line that extends to both sides till infinity and has no curves is called a straight line. The slope of the y-axis is known to be as \[\dfrac{1}{0}\]. Also, another important formula that is used in this question is as follows
\[slope=\dfrac{-coefficient\ of\ y}{coefficient\ of\ x}\].