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

& \left( 2x+3y+4 \right)+\lambda 6x-\lambda y+\lambda 12=0 \\\ & x(2+\lambda 6)+y(3-\lambda )+4+\lambda 12=0 \\\ \end{aligned}$$ Now, using the formula for calculating the slope of the given line as follows $$slope=\dfrac{\lambda -3}{2+6\lambda }$$ Now, this slope should be equal to $$\dfrac{1}{0}$$ . $$\begin{aligned} & \dfrac{\lambda -3}{2+6\lambda }=\dfrac{1}{0} \\\ & 2+6\lambda =0 \\\ & \lambda =\dfrac{-1}{3} \\\ \end{aligned}$$ Hence, this is the value of $$\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}$$.