Question

The nearest point on the line $3x - 4y = 25$ from the origin is:

Answer 1

First, we learn about the nearest point. The nearest line of the given line from the origin is the point of intersection of the given line and the perpendicular line passing through the origin.
The given line equation from the origin is $3x - 4y = 25$.
We need to find the perpendicular line using the given formulae and then we have to solve the perpendicular line and the given line to obtain the point of intersection.

Formula to be used:
The slope-intercept form is $y = mx + b$ where $m$ is the slope and $b$ is the $y$-intercept.
The formula to calculate the slope of the perpendicular line is as follows.
The slope of the perpendicular line $= \dfrac{{ - 1}}{m}$
The equation of the straight line is $y - {y_1} = m\left( {x - {x_1}} \right)$ where, $m$ is the slope and $\left( {{x_1},{y_1}} \right)$is the given point.

Complete step by step answer:
The given equation of a line is $3x - 4y = 25$………$\left( 1 \right)$
We are asked to calculate the nearest point on the given line from the origin.

We can rewrite the given equation as follows.
$3x - 4y = 25 \Rightarrow y = \dfrac{3}{4}x - \dfrac{{25}}{4}$
The above equation is in slope-intercept form.
That is, it is in the form$y = mx + b$
Where $m$ is the slope and $b$ is the $y$-intercept.
From this $y = \dfrac{3}{4}x - \dfrac{{25}}{4}$ , we get the slope of the line $m = \dfrac{3}{4}$
Now, we need to find the slope of the line perpendicular to the given line.
The slope of the perpendicular line is$\dfrac{{ - 1}}{m}$
Since we have $m = \dfrac{3}{4}$, we get
The slope of the perpendicular line $= \dfrac{{ - 1}}{m}$
$= \dfrac{{ - 4}}{3}$
Now, we shall calculate the equation of the line perpendicular to the given line drawn from the origin.
That is the given point is$(0,0)$
The equation of the straight line is $y - {y_1} = m\left( {x - {x_1}} \right)$ where, $m$ is the slope and$\left( {{x_1},{y_1}} \right)$is the given point.
Hence, the equation of the perpendicular for $3x - 4y = 25$is as follows.
$y - 0 = - \dfrac{4}{3}\left( {x - 0} \right)$
$\Rightarrow 3y = - 4x$
$\Rightarrow 4x + 3y = 0$ ………$\left( 2 \right)$
From$\left( 2 \right)$ , we have$x = - \dfrac{3}{4}y$ ………$\left( 3 \right)$
We shall substitute the value of $x$ in $\left( 1 \right)$
$3x - 4y = 25 \Rightarrow \dfrac{{ - 3}}{4}y \times 3 - 4y = 25$
$\Rightarrow \dfrac{{ - 9}}{4}y - 4y = 25$
$\Rightarrow \dfrac{{ - 9y - 16y}}{4} = 25$
$\Rightarrow \dfrac{{ - 25y}}{4} = 25$
$\Rightarrow y = 25 \times \dfrac{{ - 4}}{{25}}$
$\Rightarrow y = - 4$
We shall substitute $y = - 4$ in$\left( 3 \right)$.
$x = - \dfrac{3}{4}y$
$\Rightarrow x = - \dfrac{3}{4} \times - 4$
$\Rightarrow x = 3$
Hence, the required nearest point is $\left( {x,y} \right) = \left( {3, - 4} \right)$

Note: It can be understood that the nearest point from the origin of the line is the point where the perpendicular line and the given line intersect. The nearest point can also be known as the closest point of the line. Hence, the required nearest point is $\left( {x,y} \right) = \left( {3, - 4} \right)$