Section Formula & Conic Sections - Practice Questions & MCQ

Section Formula

1.Internal division

The coordinates of the point $\mathrm{P}(\mathrm{x}, \mathrm{y})$ dividing the line segment joining the two points $\mathrm{A}\left(\mathrm{x}_1, \mathrm{y}_1\right)$ and $\mathrm{B}\left(\mathrm{x}_2, \mathrm{y}_2\right)$ internally in the ratio m : n is given by

$
\mathbf{x}=\frac{\mathbf{m x}_{\mathbf{2}}+\mathbf{n x}_{\mathbf{1}}}{\mathbf{m}+\mathbf{n}}, \quad \mathbf{y}=\frac{\mathbf{m} \mathbf{y}_{\mathbf{2}}+\mathbf{n} \mathbf{y}_{\mathbf{1}}}{\mathbf{m}+\mathbf{n}}
$
Note:
If $P$ is the mid point of the line segment $A B$, then ratio become equals, i.e. $m=n$, in this case, coordinates of point $P$ is

$
\mathrm{x}=\frac{\mathrm{x}_1+\mathrm{x}_2}{2}, \quad \mathrm{y}=\frac{\mathrm{y}_1+\mathrm{y}_2}{2}
$

2. External Division

The coordinates of the point $P(x, y)$ dividing the line segment joining the two points $A\left(x_1, y_1\right)$ and $B\left(x_2, y_2\right)$ externally in the ratio $m: n$ is given by

$
\mathbf{x}=\frac{\mathbf{m x}_{\mathbf{2}}-\mathbf{n x}_1}{\mathbf{m}-\mathbf{n}}, \quad \mathbf{y}=\frac{\mathbf{m y}_{\mathbf{2}}-\mathbf{n y}_1}{\mathbf{m}-\mathbf{n}}
$

NOTE:
1. If the ratio in which a given line segment is divided, is to be determined, then sometimes, for convenience (instead of taking the ratio $\mathrm{m}: \mathrm{n}$ ) we take the ratio $\lambda: 1$ and apply the formula for internal division $\left(\frac{\lambda x_2+x_1}{\lambda+1}, \frac{\lambda y_2+y_1}{\lambda+1}\right)$
2. If the value of $\lambda>0$, it is an internal division, otherwise it is an external division (i.e. when $\lambda<0$ )