Question

The equation of the straight line passing through the point $\left( {3,2} \right)$ and perpendicular to the line $y = x$ is:
(A) $x - y = 5$
(B) $x + y = 5$
(C) $x + y = 1$
(D) $x - y = 1$

Answer 1

Hint : In the given problem, we are required to find the equation of a line which is perpendicular to the given line and has a given point on it. We will first find the slope of the required line using the fact that the product of two perpendicular lines is always one. We can easily find the equation of the line using the slope point form of a straight line. So, we have to substitute the slope and the point given to us in the slope and point form of the line and then simplify the equation of the straight line required.

Complete step-by-step answer :
So, we have to find the equation of a line which is perpendicular to the line $y = x$ and has point $\left( {3,2} \right)$ lying on it.
So, the slope of line $y = x$ is $1$.
Now, we know that the product of slopes of two lines that are perpendicular to each other is equal to $- 1$. Hence, the slope of the required line is $\left( { - 1} \right)$.
Also, given point on the line is $\left( {3,2} \right)$
We know the slope point form of the line, where we can find the equation of a straight line given the slope of the line and the point lying on it. The slope point form of the line can be represented as: $\left( {y - {y_1}} \right) = m\left( {x - {x_1}} \right)$ where $\left( {{x_1},{y_1}} \right)$ is the point lying on the line given to us and m is the slope of the required straight line.
Considering ${x_1} = 3$and ${y_1} = 2$ as the point given to us is $(3,2)$
Therefore, required equation of line is as follows:
$(y - 2) = - 1(x - (3))$
On opening the brackets and simplifying further, we get,
$\Rightarrow y - 2 = - x + 3$
Adding $2$ on both sides of the equation. So, we get,
$\Rightarrow y = - x + 5$
Shifting the x term to left side of the equation, we get,
$\Rightarrow y + x = 5$
Hence, the equation of the straight line is: $y + x = 5$.
So, the correct answer is “Option B”.

Note : The given problem requires us to have thorough knowledge of the concepts of coordinate geometry. The equation of the straight line in all the forms must be remembered. The applications of concepts learnt in coordinate geometry are wide ranging. We must take care of the calculations so as to be sure of the final answer.