Question

The coordinates of the foot of the perpendicular from $(a,0)$ on the line $y=mx+\frac{a}{m}$ are

• A. $\left(0,\frac{a}{m}\right)$
• B. $\left(0,\frac{-a}{m}\right)$
• C. $\left(\frac{a}{m},0\right)$
• D. $\left(\frac{-a}{m},0\right)$

Answer 1

The correct option is A

$\left(0,\frac{a}{m}\right)$

Explanation for correct option:

Given that $y=mx+\frac{a}{m}\to \left(i\right)$

Step 1:Visualizing the given information.

Slope of the line $y=mx+\frac{a}{m}$is '$m$'.

So, the slope of $AB$ will be $=-\frac{1}{m}$

Now, $AB$ passes through the points $(a,0)$.

Therefore the equation of $AB$ will be

$$\begin{gathered} y-y_1=\text{slope}\left(x-x_1\right) \\ y-0=-\frac{1}{m}\left(x-a\right) \\ \Rightarrow my=-x+a \\ \Rightarrow my+x=a\to \left(ii\right) \end{gathered}$$

Step 2:Finding the coordinates:

Solving equations $\left(i\right)$ and $\left(ii\right)$

$$\begin{gathered} \Rightarrow m^{2}x+a+x-a=0 \\ \Rightarrow x\left(m^2+1\right)=0 \\ \Rightarrow x=0 \end{gathered}$$

Substituting the value of $x$ in equation $\left(i\right)$

$$\begin{gathered} y=m\times 0+\frac{a}{m} \\ =\frac{a}{m} \end{gathered}$$

The point $B$ is $\left(0,\frac{a}{m}\right)$

Hence, option (A) is the correct answer