Determine the ratio in which the point (−6, a) divides

The co-ordinates of a point which divided two points $\left(x_{1}, y_{1}\right)$ and $\left(x_{2}, y_{2}\right)$ internally in the ratio $m: n$ is given by the formula,

$(x, y)=\left(\left(\frac{m x_{2}+n x_{1}}{m+n}\right),\left(\frac{m y_{2}+n y_{1}}{m+n}\right)\right)$

Here we are given that the point P(−6,a) divides the line joining the points A(−3,1) and B(−8,9) in some ratio.

Let us substitute these values in the earlier mentioned formula.

$(-6, a)=\left(\left(\frac{m(-8)+n(-3)}{m+n}\right),\left(\frac{m(9)+n(1)}{m+n}\right)\right)$

Equating the individual components we have

$-6=\frac{m(-8)+n(-3)}{m+n}$

$-6 m-6 n=-8 m-3 n$

$2 m=3 n$

$\frac{m}{n}=\frac{3}{2}$We see that the ratio in which the given point divides the line segment is.

Let us now use this ratio to find out the value of ‘a’.

$(-6, a)=\left(\left(\frac{m(-8)+n(-3)}{m+n}\right),\left(\frac{m(9)+n(1)}{m+n}\right)\right)$

$(-6, a)=\left(\left(\frac{3(-8)+2(-3)}{3+2}\right),\left(\frac{3(9)+2(1)}{3+2}\right)\right)$

Equating the individual components we have

$a=\frac{3(9)+2(1)}{3+2}$

$a=\frac{29}{5}$

Thus the value of ' $a$ ' is $\frac{29}{5}$.