Question

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

Find the value of $a$ and $b$.

Answer 1

Find the value of $a$ and $b$

Given that, P divides the line segment AB in the ratio of $3:1$

$A(3a+1,-2)=(x_1,y_1)$ write in the format $A(x,y)=(a,b)$

$B(8a,5)=(x_2,y_2)$

$m:n=3:1$

$m+n=4$

Section formula is used in coordinate geometry to calculate the ratio in which a line joined by two points can be divided.

Use the section formula,

$x=\frac{m{x}_2+n{x}_1}{m+n}$and $y=\frac{\left(m{y}_2+n{y}_1\right)}{\left(m+n\right)}$

Now, substitute the known values in the formula

$x=\frac{\left(3\left(8a\right)+1\left(3a+1\right)\right)}{4}$ and $y=\frac{\left(3\left(5\right)+1\left(-2\right)\right)}{4}$

$x=\frac{(24a+3a+1)}{4}$ and $y=\frac{\left(15-2\right)}{4}$

Now find the value of a:

$$\begin{gathered} \Rightarrow 9a-2=\frac{\left(27a+1\right)}{4} \\ \Rightarrow 36a-8=27a+1 \\ \Rightarrow 9a=9 \\ \Rightarrow a=1 \end{gathered}$$

Now find the value of b

$$\begin{gathered} \Rightarrow -b=\frac{13}{4} \\ \Rightarrow b=-\frac{13}{4} \end{gathered}$$

Hence the value of a is $1$ and b is $-\frac{13}{4}$.