Question

Two perpendicular lines are intersecting at (4, 3). One meeting coordinate axis at (4, 0), find the distance between the origin and the point of intersection of another line with the coordinate axis.

Answer 1

Hint: The question is related to coordinates geometry. You must know the formula of slope of line and the distance formula with the help of coordinates of two points. First find the slope and the equation of the line using the given coordinates in the question.

Complete step-by-step answer:
Given that two perpendicular lines are intersecting at (4, 3) where one line is meeting at (4, 0) with coordinate axis.
We have to find the distance between the origin and the point of intersection of another line with the coordinate axis.
First, we have to find the equation of the line of pint (4, 3) and the point (4, 0)
$y - {y_1} = m(x - {x_1})$
In the above formula m is slope of the line so, first we have to find the slope of the line using the two point
$m = \dfrac{{{y_1} - y}}{{{x_1} - x}}$
Put the values of the point
$m = \dfrac{{0 - 3}}{{4 - 4}}$
$m = \dfrac{{ - 3}}{0}$
Now we have slope of the line let’s find the equation of the line
We can choose any point for ${x_1}$ and ${y_1}$ let’s take point (4, 0)
$y - 0 = \dfrac{{ - 3}}{0}(x - 4)$
$ \Rightarrow - 3(x - 4) = 0$
Open the brackets and multiply the numbers
$ \Rightarrow - 3x + 4 = 0$
$ \Rightarrow - 3x = - 12$
$\therefore x = 4$
The equation of perpendicular line to $x = 4$ is $y = \lambda $
The perpendicular line passing through (4, 3)
Hence $\lambda = 3$ that means $y = 3$
Another perpendicular line meeting at (0, 3) with coordinates axis
So, the distance of point (0, 3) from the origin
$d = \sqrt {{{(y - {y_1})}^2} + {{(x - {x_1})}^2}} $
Putting the values of coordinates
$d = \sqrt {{{(3 - 0)}^2} + {{(0 - 0)}^2}} $
$ \Rightarrow d = \sqrt 9 $
$\therefore d = 3$

Hence, the distance between the origin and the point of intersection of another line with the coordinate axis is 3.

Note: If two lines are meeting perpendicularly then the angle between the two points is $90^\circ$. If two lines are parallel then there is no point of intersection between the two points.