Ratio Formula and Mid Point Formula

Let $${\text{P}}\left( {{{\text{x}}_1},{{\text{y}}_1}} \right)$$ and $${\text{Q}}\left( {{{\text{x}}_2},{{\text{y}}_2}} \right)$$ be any two points on the line.

Let a point $${\text{R}}\left( {{\text{x}},{\text{y}}} \right)$$ be the point which divides PQ in the ratio $${{\text{K}}_1}:{{\text{K}}_2}$$ i.e. PR : PQ = $${{\text{K}}_1}:{{\text{K}}_2}$$

From P, R and Q draw PM, RN and OL perpendicular to the X-Axis.

From ‘P’ draw $${\text{PS}} \bot $$to RN.

From ‘R’ draw $${\text{RT}} \bot $$to QL.

Since the right triangles$$\Delta {\text{PRS}}$$ and $$\Delta {\text{RQT}}$$
\[\begin{gathered} m\angle {\text{SRP}} = m\angle {\text{TRQ}} \\ m\angle {\text{PSR}} = m\angle {\text{RTQ}} \\ m\angle {\text{SRP}} = m\angle {\text{TQR}} \\ \end{gathered} \]

$$\therefore $$ $$\Delta {\text{PRS}}$$ and $$\Delta {\text{RQT}}$$ are similar triangles.

\[\begin{gathered} {\text{PS}}:{\text{RT}} = {\text{RS}}:{\text{QT}} = {\text{PR}}:{\text{RQ}} \\ \frac{{{\text{PS}}}}{{{\text{RT}}}} = \frac{{{\text{RS}}}}{{{\text{QT}}}} = \frac{{{\text{PR}}}}{{{\text{RQ}}}} \\ \frac{{{\text{MN}}}}{{{\text{NL}}}} = \frac{{{\text{RN}} – {\text{SN}}}}{{{\text{QL}} – {\text{TL}}}} = \frac{{{{\text{K}}_{\text{1}}}}}{{{{\text{K}}_{\text{2}}}}} \\ \frac{{{\text{ON}} – {\text{OM}}}}{{{\text{OL}} – {\text{ON}}}} = \frac{{{\text{RN}} – {\text{SN}}}}{{{\text{QL}} – {\text{TL}}}} = \frac{{{{\text{K}}_{\text{1}}}}}{{{{\text{K}}_{\text{2}}}}} \\ \frac{{{\text{x}} – {{\text{x}}_{\text{1}}}}}{{{{\text{x}}_{\text{2}}} – {\text{x}}}} = \frac{{{\text{y}} – {{\text{y}}_{\text{1}}}}}{{{{\text{y}}_{\text{2}}} – {\text{y}}}} = \frac{{{{\text{K}}_{\text{1}}}}}{{{{\text{K}}_{\text{2}}}}} \\ \end{gathered}\]

\[\frac{{{\text{x}} – {{\text{x}}_{\text{1}}}}}{{{{\text{x}}_{\text{2}}} – {\text{x}}}} = \frac{{{{\text{K}}_{\text{1}}}}}{{{{\text{K}}_{\text{2}}}}}\]

Cross Multiplication
\[\begin{gathered} {{\text{K}}_{\text{2}}}\left( {{\text{x}} – {{\text{x}}_{\text{1}}}} \right) = {{\text{K}}_{\text{1}}}\left( {{{\text{x}}_{\text{2}}} – {\text{x}}} \right) \\ {{\text{K}}_{\text{2}}}{\text{x}} – {{\text{K}}_{\text{2}}}{{\text{x}}_{\text{1}}} = {{\text{K}}_{\text{1}}}{{\text{x}}_{\text{2}}} – {{\text{K}}_{\text{1}}}{\text{x}} \\ {{\text{K}}_{\text{1}}}{\text{x}} + {{\text{K}}_{\text{2}}}{\text{x}} = {{\text{K}}_{\text{1}}}{{\text{x}}_{\text{2}}} + {{\text{K}}_{\text{2}}}{{\text{x}}_{\text{1}}} \\ \left( {{{\text{K}}_{\text{1}}} + {{\text{K}}_{\text{2}}}} \right){\text{x}} = {{\text{K}}_{\text{1}}}{{\text{x}}_{\text{2}}} + {{\text{K}}_{\text{2}}}{{\text{x}}_{\text{1}}} \\ {\text{x}} = \frac{{{{\text{K}}_{\text{1}}}{{\text{x}}_{\text{2}}} + {{\text{K}}_{\text{2}}}{{\text{x}}_{\text{1}}}}}{{{{\text{K}}_{\text{1}}} + {{\text{K}}_{\text{2}}}}} \\ \end{gathered} \]

\[\frac{{{\text{y}} – {{\text{y}}_{\text{1}}}}}{{{{\text{y}}_{\text{2}}} – {\text{y}}}} = \frac{{{{\text{K}}_{\text{1}}}}}{{{{\text{K}}_{\text{2}}}}}\]

Cross Multiplication
\[\begin{gathered} {{\text{K}}_{\text{2}}}\left( {{\text{y}} – {{\text{y}}_{\text{1}}}} \right) = {{\text{K}}_{\text{1}}}\left( {{{\text{y}}_{\text{2}}} – {\text{y}}} \right) \\ {{\text{K}}_{\text{2}}}{\text{y}} – {{\text{K}}_{\text{2}}}{{\text{y}}_{\text{1}}} = {{\text{K}}_{\text{1}}}{{\text{y}}_{\text{2}}} – {{\text{K}}_{\text{1}}}{\text{y}} \\ {{\text{K}}_{\text{1}}}{\text{y}} + {{\text{K}}_{\text{2}}}{\text{y}} = {{\text{K}}_{\text{1}}}{{\text{y}}_{\text{2}}} + {{\text{K}}_{\text{2}}}{{\text{y}}_{\text{1}}} \\ \left( {{{\text{K}}_{\text{1}}} + {{\text{K}}_{\text{2}}}} \right){\text{y}} = {{\text{K}}_{\text{1}}}{{\text{y}}_{\text{2}}} + {{\text{K}}_{\text{2}}}{{\text{y}}_{\text{1}}} \\ {\text{y}} = \frac{{{{\text{K}}_{\text{1}}}{{\text{y}}_{\text{2}}} + {{\text{K}}_{\text{2}}}{{\text{y}}_{\text{1}}}}}{{{{\text{K}}_{\text{1}}} + {{\text{K}}_{\text{2}}}}} \\ \end{gathered} \]

Required Point
\[\boxed{{\text{R}}\left( {{\text{x}},{\text{y}}} \right) = \left( {\frac{{{{\text{K}}_{\text{1}}}{{\text{x}}_{\text{2}}} + {{\text{K}}_{\text{2}}}{{\text{x}}_{\text{1}}}}}{{{{\text{K}}_{\text{1}}} + {{\text{K}}_{\text{2}}}}},\frac{{{{\text{K}}_{\text{1}}}{{\text{y}}_{\text{2}}} + {{\text{K}}_{\text{2}}}{{\text{y}}_{\text{1}}}}}{{{{\text{K}}_{\text{1}}} + {{\text{K}}_{\text{2}}}}}} \right)}\]