Question

Find the length of AB if the line 3x $-$ 4y + 12 = 0 intersects the X -axis and Y-axis at point A and point B respectively.

Answer 1

The equation of the line is 3x $-$ 4y + 12 = 0.
It is given that the line intersects the X-axis at point A and the y-coordinate is 0 on the X-axis.
To find the x-coordinate, put y = 0 in the above equation.
i.e., 3x $-$ 4(0) + 12 = 0
⇒ 3x = $-$12
⇒ x = $\frac{-12}{3}=-4$
So, the coordinates of A are ($-$4,0).

Similarly, it is given that the line intersects the Y-axis at point B and the x-coordinate is 0 on the Y-axis.
To find the y-coordinate, put x = 0 in the above equation.

i.e., 3(0) $-$ 4y + 12 = 0
⇒ $-$4y = $-$12
⇒ y = $\frac{-12}{-4}=3$
So, the coordinates of B are (0,3).
Also, if O is the origin then, OA = 4 units and OB = 3 units.
Since AOB is a right angled triangle, by Pythagoras theorem, we have:

$$\begin{gathered} {\mathrm{AB}}^2={\mathrm{AO}}^2+{\mathrm{BO}}^2 \\ =4^2+3^2 \\ =16+9 \\ =25 \\ =5\mathrm{units} \end{gathered}$$