Question

a point p(6,0) divides line segment ab in a ratio of 2:1 from endpoint b(4,2). determine the coordinates of point a.

Explanation:

Step1: Use section - formula for x - coordinate

Let the coordinates of point $A$ be $(x,y)$. The section - formula when a point $P(x_p,y_p)$ divides the line segment joining $A(x_1,y_1)$ and $B(x_2,y_2)$ in the ratio $m:n$ is $x_p=\frac{mx_2+nx_1}{m + n}$ and $y_p=\frac{my_2+ny_1}{m + n}$. Here, $P(6,0)$, $B(4,2)$, $m = 2$, $n = 1$. For the x - coordinate:
$6=\frac{2\times4 + 1\times x}{2 + 1}$

Step2: Solve for x

Cross - multiply the equation $6=\frac{8 + x}{3}$:
$6\times3=8 + x$
$18=8 + x$
$x=18 - 8=10$

Step3: Use section - formula for y - coordinate

For the y - coordinate:
$0=\frac{2\times2+1\times y}{2 + 1}$

Step4: Solve for y

Cross - multiply the equation $0=\frac{4 + y}{3}$:
$0\times3=4 + y$
$y=-4$

Answer:

$(10,-4)$