Ex.7.4 Q1 Coordinate Geometry Solution - NCERT Maths Class 10

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Question

Determine the ratio in which the line\(\begin{align}2x + y - {\text{ }}4{\text{ }} = {\text{ }}0\end{align}\) divides the line segment joining the points \(\begin{align} A\left( {2,\,\, - {\text{ }}2} \right){\text{ and }}{\text{ }}B \left( {3,\,\,7} \right)\end{align}\).

Text Solution

 

Reasoning:

The coordinates of the point\(\;{P}\,(x, y)\) which divides the line segment joining the points \({A(x1, y1) \;\text{and}\; B(x_2, y_2)}\), internally, in the ratio \(\rm{m1\,:\,m2}\) is given by the Section Formula.

\(\begin{align}{{P}}({\text{x}},{\text{y}}) = \left[ {\frac{{{\text{m}}{{\text{x}}_2} + {\text{n}}{{\text{x}}_1}}}{{{\text{m}} + {\text{n}}}},\frac{{{\text{m}}{{\text{y}}_2} + {\text{n}}{{\text{y}}_1}}}{{{\text{m}} + {\text{n}}}}} \right] & & \end{align}\)

What is known?

The \(x\) and \(y\) co-ordinates of the points \(A\) and \(B\).

The equation of the straight line i.e. \(\begin{align}2x + y - {\text{ }}4{\text{ }} = {\text{ }}0\end{align}\)

What is unknown?

The ratio in which the line \(\begin{align}2x + y - {\text{ }}4{\text{ }} = {\text{ }}0\end{align}\) divides the line segment joining the points \(\begin{align}{\text{ A}}\left( {2,\,\, - {\text{ }}2} \right){\text{ }}{\text{ and}}{\text{ B}}\left( {3,\,\,7} \right)\end{align}\)

Steps:

From the Figure,

Given,

  • Let the given line \(\begin{align}2x + y - 4{\text{ }} = 0\end{align}\) divide the line segment joining the points \(\begin{align}A\left( {2,{\text{ }} - 2} \right){\text{ }}{\text{and }}B\left( {3,{\text{ }}7} \right)\end{align}\) in a ratio \(k : 1\) at point \(C\).

Coordinates of the point of division, \(\begin{align}\text{C}(x,y) = \left( {\frac{{3k + 2}}{{k + 1}},\,\,\frac{{7k - 2}}{{k + 1}}} \right)\end{align}\)

This point \(C\) also lies on \(\begin{align}2x + y - {\text{ }}4{\text{ }} = {\text{ }}0 & & \ldots \ldots \ldots {\text{Equation }}\left( 1 \right)\end{align}\)

By substituting the values of \(\begin{align}\text{C}\left( {x,{\text{ }}y} \right)\end{align}\) in Equation (1),

\(\begin{align}   ∴\; 2\left( {\frac{{3k + 2}}{{k + 1}}} \right) + \left( {\frac{{7k - 2}}{{k + 1}}} \right) - 4 = 0 \end{align}\)

\(\begin{align}   \frac{{6k + 4 + 7k - 2 - 4k - 4}}{{k + 1}} = 0 \qquad \quad  \end{align}\) (By Cross multiplying & transposing) 

\(\qquad\qquad\begin{align}  9k - 2 = 0 \end{align}\)

\(\qquad\qquad\begin{align}k = \frac{2}{9}\end{align}\)

Therefore, the ratio in which the line \(\begin{align}2x + y - {\text{ }}4{\text{ }} = {\text{ }}0\end{align}\) divides the line segment joining the points \(\begin{align}{{A }}\left( {2,{\text{ }} - 2} \right){\text{ }}{\text{ and}}\;{{B }}\left( {3,{\text{ }}7} \right)\end{align}\) is \(2:9\) internally.

  
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