Question

The coordinates of the foot of the perpendicular drawn from the point $(3,4)$ on the line $2x+y-7=0$ is

• A. $\left(\frac{9}{5},\frac{17}{5}\right)$
• B. $(1,5)$
• C. $(-5,1)$
• D. $(1,-5)$

Answer 1

The correct option is A

$\left(\frac{9}{5},\frac{17}{5}\right)$

Explanation for the correct option:

Foot of perpendicular:

Given the equation,

$$\begin{gathered} 2x+y-7=0 \\ \Rightarrow y=-2x+7 \end{gathered}$$

Therefore slope $=-2$

So, the slope of $AB=\frac{1}{2}$ , since the product of slopes of two perpendicular lines is equal to $-1$.

Since $AB$ passes through $(3,4)$

Therefore, the equation of $AB$ will be:

$$\begin{gathered} \Rightarrow y-4=\frac{1}{2}(x-3) \\ \Rightarrow 2y-x-5=0 \end{gathered}$$

Finding the point of intersection of these two lines to find the point $B$

Solving $2x+y=0\&\left(2y-x-5=0\right)\times 2$

$$\begin{gathered} \Rightarrow \frac{\begin{array}{l}2x+y=7\\ -2x+4y=10\end{array}}{5y=17\Rightarrow y={\displaystyle \frac{17}{5}}} \\ \therefore x=2y-5 \\ =\frac{34}{5}-5 \\ \Rightarrow x=\frac{9}{5} \end{gathered}$$

Thus the coordinates of $B$ is $\left(\frac{9}{5},\frac{17}{5}\right)$

Hence, option A is the correct answer.