Coordinate Geometry

**• Section formula:**

The co-ordinates 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*:

*n*, are given by:

$\mathrm{P}\left(x,y\right)=\left(\frac{m{x}_{2}+n{x}_{1}}{m+n},\frac{m{y}_{2}+n{y}_{1}}{m+n}\right)$

**Example:**In what ratio does the point (–4, 7) divide the line segment joining the points P (–1, 1) and Q (–6, 11).

**Solution:**Let the point (–4, 7) divide the line segment joining the points P (–1, 1) and Q(–6, 11) in the ratio λ : 1.

Thus, by section formula, we have:

$\left(\frac{-6\lambda +\left(-1\right)}{\lambda +1},\frac{11\lambda +1}{\lambda +1}\right)=\left(-4,7\right)\phantom{\rule{0ex}{0ex}}\Rightarrow \frac{-6\lambda -1}{\lambda +1}=-4,\frac{11\lambda +1}{\lambda +1}=7\phantom{\rule{0ex}{0ex}}\Rightarrow -6\lambda -1=-4\lambda -4\phantom{\rule{0ex}{0ex}}\Rightarrow 2\lambda =3\phantom{\rule{0ex}{0ex}}\Rightarrow \lambda =\frac{3}{2}$

Therefore, the required ratio is 3:2.

• The

**mid-point**of the line segment join…

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