Question

If a line passing through $(1,1)$ divides the segment between the axes in the ratio 3:4, then the equation of the line is

• A. $4x+3y=7$
• B. $4x-3y=-7$
• C. $4x-3y=7$
• D. $4x+3y=-7$

Answer 1

The correct option is A

$4x+3y=7$

Let the line through $P(1,1)$ cuts the $x$-axis at $A(a,0)$ and $y$-axis at $B(0,b)$.

We know that the coordinates of the point $P(x,y)$ dividing the join of $A(x_1,y_1)$ and $B(x_2,y_2)$in the ratio $m:n$ internally is given by

$x=\frac{m{x}_2+n{x}_1}{m+n}$ and $y=\frac{m{y}_2+n{y}_1}{m+n}$

Given AP:BP=3:4

Then, $1=\frac{3\times 0+4\times a}{3+4}⟹4a=7⟹a=\frac{7}{4}$

Also, $1=\frac{3\times b+4\times 0}{3+4}⟹3b=7⟹b=\frac{7}{3}$

We know that, slope(m) of the line formed by joining the points $(x_1,y_1)$ and $(x_2,y_2)$ is given by

$m=\frac{y_2-y_1}{x_2-x_1}$

Then slope of AB = $\frac{\frac{7}{3}-0}{0-\frac{7}{4}}=\frac{-4}{3}$

We know that equation of line through point $(x_1,y_1)$ is given by,

$y-y_1=m(x-x_1)$

Then equation of AB is

$y-1=\frac{-4}{3}(x-1)$

$⟹3y-3=-4x+4$

$⟹4x+3y-7=0$

i.e. $4x+3y=7$