Question

The coordinates of the foot of the perpendicular drawn from $(2,4)$ to the line $x+y=1$ is

• A. $\left(\frac{1}{3},\frac{3}{2}\right)$
• B. $\left(-\frac{1}{2},\frac{3}{2}\right)$
• C. $\left(\frac{4}{3},\frac{1}{2}\right)$
• D. $\left(\frac{3}{4},-\frac{1}{2}\right)$

Answer 1

The correct option is B

$\left(-\frac{1}{2},\frac{3}{2}\right)$

Explanation for the correct option:

Step:1 Finding slope of line AB

Visualizing the given information,

Given the equation:

$$\begin{gathered} x+y=1 \\ \Rightarrow y=1-x \end{gathered}$$

Comparing with $y=mx+c$ Therefore, slope of this line is $m=-1$

Using perpendicular lines slope formula $m_1\times m_2=-1$ Therefore the slope of $AB$ as it is perpendicular to this line $=-\frac{1}{-1}=1$

Step:2 Finding equation of line AB

$AB$ passes through the point $(2,4)$

Therefore the equation of $AB$

$$\begin{gathered} \Rightarrow y-4=1\left(x-2\right) \\ \Rightarrow x-y+2=0 \end{gathered}$$

Step:3 Finding the point of intersection of $AB\&CD$

$$\begin{gathered} \Rightarrow x+y=1\&x-y+2=0 \\ \Rightarrow 2x=-1 \\ \Rightarrow x=-\frac{1}{2} \end{gathered}$$

Putting value of $x$, we have,

$$\begin{gathered} y=1-x \\ y=1+\frac{1}{2} \\ y=\frac{3}{2} \end{gathered}$$

Therefore, coordinates of $B$ is $\left(-\frac{1}{2},\frac{3}{2}\right)$

Hence, option (B) is the correct answer.