Question

The coordinates of the foot of the perpendicular drawn from the point $\left( {2,4} \right)$ to the line $x + y = 1$ is:
(A) $\left( {\dfrac{1}{3},\dfrac{3}{2}} \right)$
(B) $\left( { - \dfrac{1}{2},\dfrac{3}{2}} \right)$
(C) $\left( {\dfrac{4}{3},\dfrac{1}{2}} \right)$
(D) $\left( {\dfrac{3}{4}, - \dfrac{1}{2}} \right)$

Answer 1

Hint: In the given problem, we are required to find the coordinates of the foot perpendicular to the line $x + y = 1$ from the point $\left( {2,4} \right)$. We first find the slope of the line given to us, $x + y = 1$. Then, we find the slope of the required line. Then, we use the slope point form of a straight line in order to get to the equation of the required line.

Complete step-by-step answer:
So, we are provided with the equation of a straight line as $x + y = 1$.
We find the slope of this straight line by resembling the equation of line with the slope intercept form of a straight line.
We know the slope intercept form a line as $y = mx + c$ where m is the slope of the straight line and c is the y intercept. So, keeping the y term on the left side of equation and shifting the x term to right side of equation, we get,
$ \Rightarrow y = - x + 1$
So, the slope of the given line is $\left( { - 1} \right)$ comparing to the slope intercept form of a line. Now, we know that the product of slopes of perpendicular line is always equal to $ - 1$. So, the slope of any line perpendicular to the given line will be $1$ as $1 \times \left( { - 1} \right) = - 1$.
So, we get the slope of required line as $1$. Now, we also have the coordinates of a point lying in the line as $\left( {2,4} \right)$. 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 ${x_1} = 2$ and ${y_1} = 4$ as the coordinates of point lying on the line are $\left( {2,4} \right)$.
Therefore, required equation of line is as follows:
$\left( {y - 4} \right) = 1\left( {x - 2} \right)$
On opening the brackets and simplifying further, we get,
$ \Rightarrow y - 4 = x - 2$
Now, shifting the terms in the equation, we get,
 $ \Rightarrow y = x + 2$
Now, we have to find the intersection of the two perpendicular lines $y = x + 2$ and $x + y = 1$.
Substituting the value of x from the equation $y = x + 2$ into the equation $x + y = 1$, we get,
$ \Rightarrow x + \left( {x + 2} \right) = 1$
Simplifying the equation, we get,
$ \Rightarrow 2x = - 1$
Dividing both sides of the equation by $2$, we get,
$ \Rightarrow x = \dfrac{{ - 1}}{2}$
Now, putting in the value of x in the equation $x + y = 1$, we get,
$ \Rightarrow \left( { - \dfrac{1}{2}} \right) + y = 1$
Finding value of y by shifting the terms, we get,
$ \Rightarrow y = \dfrac{3}{2}$
So, we get the coordinates of the foot perpendicular as $\left( { - \dfrac{1}{2},\dfrac{3}{2}} \right)$.
Hence, option (B) is the correct answer.
So, the correct answer is “Option B”.

Note: The given problem requires us to have thorough knowledge of the concepts of coordinate geometry. We should remember the slope point 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 should take care of the calculations while doing such problems.