Question

A straight line passes through the points $(5,0)$and $(0,3)$. The length of the perpendicular from the point $(4,4)$ on the line is

• A. $\frac{15}{\sqrt{34}}$
• B. $\frac{\sqrt{17}}{2}$
• C. $\frac{17}{2}$
• D. $\sqrt{\left(\frac{17}{2}\right)}$

Answer 1

The correct option is D

$\sqrt{\left(\frac{17}{2}\right)}$

Explanation for the correct option:

Step 1: Finding the equation of line.

The points $(x_1,y_1)and(x_2,y_2)$are $(5,0)and(0,3)$

The equation of line passes through two points $(x_1,y_1)$ and $(x_2,y_2)$ is

$y-y_1=m(x-x_1)$, where $m=\frac{y_2-y_1}{x_{2-}{x}_1}$

$$\begin{gathered} \Rightarrow y-0=\left(\frac{3-0}{0-5}\right)(x-5) \\ \Rightarrow y=-\frac{3}{5}(x-5) \\ \Rightarrow 5y=-3x+15 \\ \Rightarrow 3x+5y-15=0 \end{gathered}$$

Step 2: Finding the distance:

The distance from the point $(x_1,y_1)$ to a line $ax+by+c=0$ is

$d=\left|\frac{\left(a{x}_1+b{y}_1+c\right)}{\sqrt{\left(a^2+b^2\right)}}\right|$

Thus the length of the perpendicular from the point $(4,4)$ to the line $3x+5y-15=0$ is

$$\begin{gathered} \Rightarrow d=\left|\frac{\left(3\right)\left(4\right)+\left(5\right)\left(4\right)-15}{\sqrt{3^2+5^2}}\right| \\ =\left|\frac{12+20-15}{\sqrt{9+25}}\right| \\ =\left|\frac{17}{\sqrt{34}}\right| \\ =\left|\frac{\sqrt{17}\times \sqrt{17}}{\sqrt{17}\times \sqrt{2}}\right| \\ =\sqrt{\frac{17}{2}} \end{gathered}$$

Hence, option (D) $\sqrt{\frac{\mathbf{17}}{\mathbf{2}}}$ is the correct answer.