Question

A line is such that its segment between the axes is bisected at the point $(23,27),$ the product of the intercepts made by the line on the coordinate axes is

Answer 1

Let intersection points of the line with the $x$-axis and $y$-axis are $A(a,0)$ and $B(0,b)$ respectively.

Let the midpoint of the segment be $P(23,27)$

Since, $P$ is the midpoint of $A$ and $B$

$P$ divides $AB$ in the ratio $1:1$

By Midpoint Formula,

$\Rightarrow (23,27)=(\frac{a+0}{2},\frac{0+b}{2})$

Now,$\frac{a+0}{2}=23$

$\Rightarrow \frac{a}{2}=23$

$\Rightarrow a=46$

and, also$\frac{0+b}{2}=27$

$\Rightarrow \frac{b}{2}=27$

$\Rightarrow b=54$

So, the intercepts in the axes are $46$ and $54$ respectively.

Product of intercepts $=46\times 54$

$=2484$

Hence, the required answer is $2484$ .