Question

The endpoints of $AB$ are $A( - 4,8)$ and $B(12, - 4)$ . How do you find the coordinates of $P$ if $P$ lies on $AB$ and is $\dfrac{3}{8}$ the distance from $A$ to $B$ ?

Answer 1

Hint: For solving this particular problem we will use section formula and here point $P$ internally divides the line$AB$ with endpoints $A( - 4,8) \equiv ({x_1},{y_1})\& B(12, - 4) \equiv ({x_2},{y_2})$ . The coordinates of point $P$ are given by internal division formula as follows ,
$\left( {\dfrac{{m{x_2} + n{x_1}}}{{m + n}},\dfrac{{m{y_2} + n{y_1}}}{{m + n}}} \right)$ .

Complete step by step answer:
Assuming the point $P$ lies on the line $AB$ specified
It is given that ,
$PA:PB = 3:8 \equiv m:n$
point $P$ internally divides the line$AB$ with endpoints $A( - 4,8) \equiv ({x_1},{y_1})\& B(12, - 4) \equiv ({x_2},{y_2})$ .
The coordinates of point $P$ are given by internal division formula as follows ,
$\left( {\dfrac{{m{x_2} + n{x_1}}}{{m + n}},\dfrac{{m{y_2} + n{y_1}}}{{m + n}}} \right)$ (here we are using section formula.)
$ \equiv \left( {\dfrac{{3.12 + 8( - 4)}}{{3 + 8}},\dfrac{{3( - 4) + 8(8)}}{{3 + 8}}} \right)$
$ \equiv \left( {\dfrac{4}{{11}},\dfrac{{52}}{{11}}} \right)$

Additional Information:
When to some extent $C$ divides a segment $AB$ within the ratio $m:n$ , we use the section formula to seek out the coordinates of that time. The section formula has $2$types. These types depend upon the position of point $C$. It is often present between the two points or outside the segment.
The two types are:
Internal Section Formula
External Section Formula
The coordinates of a degree $P$ which divides the line joining two points $A({x_1},{y_1},{z_1})$ and $B({x_2},{y_2},{z_2})$ internally within the ratio of $m:n$ is ,
$\left( {\dfrac{{m{x_2} + n{x_1}}}{{m + n}},\dfrac{{m{y_2} + n{y_1}}}{{m + n}},\dfrac{{m{z_2} + n{z_1}}}{{m + n}}} \right)$
The coordinates of a degree $P$ which divides the line joining two points $A({x_1},{y_1},{z_1})$ and $B({x_2},{y_2},{z_2})$ externally within the ratio of $m:n$ is ,
$\left( {\dfrac{{m{x_2} - n{x_1}}}{{m - n}},\dfrac{{m{y_2} - n{y_1}}}{{m - n}},\dfrac{{m{z_2} - n{z_1}}}{{m - n}}} \right)$

Note: The coordinates of a degree $P$ which divides the line joining two points $A({x_1},{y_1},{z_1})$ and $B({x_2},{y_2},{z_2})$ internally within the ratio of $m:n$ is ,
$\left( {\dfrac{{m{x_2} + n{x_1}}}{{m + n}},\dfrac{{m{y_2} + n{y_1}}}{{m + n}},\dfrac{{m{z_2} + n{z_1}}}{{m + n}}} \right)$ . $P$ divides a segment $AB$ within the ratio $m:n$ , we use the section formula to seek out the coordinates of that time.