Question

Find the coordinates of the points at which line joining $(2,3)$and $(3,2)$ meet the $x-$axis and $y-$axis respectively

Answer 1

Find the equation of the line joining the given points and thus find its intercepts

Let $(2,3)=\left(x_1,y_1\right)$ and $(3,2)=\left(x_2,y_2\right)$

Using two point form of equation of a line

$\frac{y-y_1}{y_2-y_1}=\frac{x-x_1}{x_2-x_1}$

$\Rightarrow$$\frac{y-3}{2-3}=\frac{x-2}{3-2}$

$\Rightarrow$$-\left(y-3\right)=x-2$

$\Rightarrow$$x+y-5=0$ $...\left(i\right)$

$x+y-5=0$ is the equation of the line joining $(2,3)$ and $(3,2)$

Now, we write this equation in double intercept form

$\frac{x}{a}+\frac{y}{b}=1$ where $a$ is the $x-$intercept and $b$ is the $y-$intercept

$x+y=5$ …[From(i)]

Dividing by $5$ throughout the equation we get

$\Rightarrow$ $\frac{x+y}{5}=1$

$\Rightarrow$$\frac{x}{5}+\frac{y}{5}=1$

Hence, $x-$intercept$=a=5$

$y-$intercept$=b=5$

Hence the co-ordinates of the points where the line intersects the axes are

$$\begin{gathered} (x,0)=(5,0) \\ (y,0)=(0,5) \end{gathered}$$

Hence, the line joining $(2,3)$and $(3,2)$ cuts the $x-$axis at $\left(5,0\right)$ and the $y-$axis at $\left(0,5\right)$ respectively.