Question

The line $x+y=10$ divides line segment $AB$ in the ratio $a:1$. Find the value of $a$.

• A. $3$
• B. $2$
• C. $1$
• D. $\frac{1}{2}$

Answer 1

The correct option is C

$1$

The point, say $P(x,y)$, divides the line $AB$ into the ratio $a:1$. The equation for the point that divides a line in the ratio m : n is,
$(\frac{(n\times x_1+m\times x_2)}{m+n},\frac{n\times y_1+m\times y_2}{m+n})$
Where $(x_1,y_1)and(x_2,y_2)$ are the coordinates of the endpoints of the line segment.
Applying the formula, we get $\frac{(1\times 2+a\times 6)}{a+1},\frac{1\times 4+a\times 8}{a+1})$
This point lies on the line $x+y=10$, so substitute the points in the equation for the line.
$\frac{(1\times 2+a\times 6)}{a+1}+\frac{1\times 4+a\times 8}{a+1}=106a+2+8a+4=10(a+1)\Rightarrow 4a=4\Rightarrow a=1$