Question

If $P$ $(9a-2,b)$ divides line segment joining $A$ $(3a+1,-3)$ and $B$ $(8a,5)$ in the ratio $3:1$, find the values of $a$ and $b.$

• A. $a=1$, $b=-1$
• B. $a=1$, $b=3$
• C. $a=\frac{11}{19}$, $b=3$
• D. $a=\frac{11}{19}$, $b=-1$
  

Answer 1

The correct option is B $a=1$, $b=3$

Note:

If a point $P(x,y)$ divides the line segment $AB$ joining the points $A(x_1,y_1)\&B(x_2,y_2)$ in the ratio $m:n$ then, by the section formula,

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

Let the point $P(x,y)=(9a-2,b)$ divide the line segment $AB$ joining the points $A(x_1,y_1)=(3a+1,-3)\&B(x_2,y_2)=(8a,5)$ in the ratio $m:n=3:1.$

Then, by the section formula, $x=\frac{n{x}_2+m{x}_1}{m+n}$

$\Rightarrow 9a-2=\frac{3\times 8a+1\times (3a+1)}{3+1}$

$\Rightarrow a=1$

Again, $y=\frac{n{y}_1+m{y}_2}{m+n}$

$\Rightarrow b=\frac{3\times 5+1\times (-3)}{3+1}=3$

So, $a=1$ and $b=3$