Question

If a line with y intercept $2$ is perpendicular to the line $3x - 2y = 6$, then its x intercept is:
(A) $1$
(B) $2$
(C) $ - 4$
(D) $3$

Answer 1

Hint: In the given problem, we are required to find the intercept of an equation which is perpendicular to a line whose equation is given to us. We are also given the y intercept of the required line. So, we first find the slope of the given line. Then, we find the slope of the required line. Then, we will substitute the values of known quantities in the slope point form of a straight line to get to the final answer.

Complete step-by-step answer:
So, we are given the equation of a line as $3x - 2y = 6$. We first calculate the slope of this line by resembling it to the slope and intercept form of a straight line $y = mx + c$.
So, we shift the y term to the right side of the equation. So, we get,
$ \Rightarrow 3x = 6 + 2y$
Shifting the constant term to the left side of the equation and dividing both sides of the equation by two. So, we get,
$ \Rightarrow y = \left( {\dfrac{{3x - 6}}{2}} \right)$
Separating the denominators, we get,
$ \Rightarrow y = \dfrac{3}{2}x - 3$
So, the slope of the straight line $3x - 2y = 6$ is $\dfrac{3}{2}$.
Now, we know that the product of the slopes of two perpendicular lines is always $ - 1$. Since the slope of the given line is $\dfrac{3}{2}$, so the slope of any line perpendicular to the given line is $ - \dfrac{2}{3}$ since $\dfrac{3}{2} \times \left( { - \dfrac{2}{3}} \right) = - 1$.
Now, we have the slope of the required line as $ - \dfrac{2}{3}$. Also, the y-intercept of the required line is given as $2$.
We know the slope intercept form of the line, where we can find the equation of a straight line given the slope of the line and the y intercept of a line. The slope intercept form of the line can be represented as: $y = mx + c$ where $c$ is the y intercept of the straight line and m is the slope of the straight line.
So, we get the equation of the straight line as, $y = \left( { - \dfrac{2}{3}} \right)x + 2$.
Now, we have to find the x intercept of $y = \left( { - \dfrac{2}{3}} \right)x + 2$. So, we know that the ordinate of any point lying on the x axis is zero. So, we substitute $y = 0$ in the equation to find the x intercept.
So, we get,
$ \Rightarrow 0 = \left( { - \dfrac{2}{3}} \right)x + 2$
Shifting the terms, we get,
$ \Rightarrow \dfrac{2}{3}x = 2$
Cross multiplying the terms, we get,
$ \Rightarrow x = 3$
So, we get the y intercept of the straight line as $3$.
So, the correct answer is “Option D”.

Note: The given problem requires us to have thorough knowledge of the concepts of coordinate geometry. We should remember the slope intercept form of a straight line to get the required answer. One must know that the product of the slopes of two perpendicular lines is one. We must take care of the calculations in order to be sure of the final answer.