Question

The coordinates of the point P(x, y) which divides the line segment joining the points $A(x_1,y_1)$ and $B(x_2,y_2)$ internally in the ratio $m_1:m_2$ are

$(\frac{m_1{x}_2-m_2{x}_1}{m_1+m_{2}1},\frac{m_1{y}_2-m_2{y}_1}{m_1+m_2})$

• A. True
• B. False
• C. Ambiguous
• D. Data insufficient

Answer 1

The correct option is B False

Let $A(x_1,y_1)$ and $B(x_2,y_2)$ be two distinct points such that a point P(x, y) divides AB internally in the ratio $(m_1:m_2)$. (ie) $\frac{AP}{PB}$ = $\frac{m_1}{m_2}$
From the figure we know that \triangle AFP and \triangle PGB are similar.
$\frac{AF}{PG}$ = $\frac{PF}{BG}$ = $\frac{AP}{PB}$= $\frac{m_1}{m_2}$
Now take any two fractions,
$\frac{AF}{PG}$ = $\frac{m_1}{m_2}$
$\Rightarrow$ $\frac{x-x_1}{x_2-x}$ = $\frac{m_1}{m_2}$
Doing Cross multiplication, x = $\frac{m_1{x}_2+m_2{x}_1}{m_1+m_2}$
Then,
$\frac{PF}{BG}$ = $\frac{AP}{PB}$= $\frac{m_1}{m_2}$
$\Rightarrow$ $\frac{y-y_1}{y_2-y}$ = $\frac{m_1}{m_2}$
Doing Cross multiplication, y = $\frac{m_1{y}_2+m_2{y}_1}{m_1+m_2}$