Ex.7.4 Q2 Coordinate Geometry Solution - NCERT Maths Class 10

Find a relation between \(x\) and \(y\) if the points \(\begin{align}\rm\left( {x,{\text{ }}y} \right),{\text{ }}\left( {1,{\text{ }}2} \right){\text{ }}and{\text{ }}\left( {7,{\text{ }}0} \right)\end{align}\) are collinear.

Text Solution

Reasoning:

Three or more points are said to be collinear if they lie on a single straight line .

What is known?

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

What is unknown?

The relation between \(x\) and \(y\) if the points \(\begin{align}\left( {x,{\text{ }}y} \right),{\text{ }}\left( {1,{\text{ }}2} \right){\text{and}}{\text{ }}\left( {7,{\text{ }}0} \right)\end{align}\) are collinear.

Steps:

Given,

\[\begin{align}{ \bullet {\text{ Let A}}\left( {x1,{\text{ }}y1} \right){\text{ }} = {\text{ }}\left( {x,{\text{ }}y} \right)}\\{ \bullet {\text{Let B}}\left( {x2,{\text{ }}y2} \right){\text{ }} = {\text{ }}\left( {1{\text{ }},{\text{ }}2} \right)}\\{ \bullet {\text{ Let C}}\left( {x3,{\text{ }}y3} \right){\text{ }} = {\text{ }}\left( {7,{\text{ }}0} \right)}\end{align}\]

If the given points are collinear, then the area of triangle formed by these points will be \(0\).

Area of a triangle \(\begin{align} = \frac{1}{2}\left\{ {{{\text{x}}_1}\left( {{{\text{y}}_2} - {{\text{y}}_3}} \right) + {{\text{x}}_2}\left( {{{\text{y}}_3} - {{\text{y}}_1}} \right) + {{\text{x}}_3}\left( {{{\text{y}}_1} - {{\text{y}}_2}} \right)} \right\} & \ldots \ldots \end{align}\) Equation (1)

By substituting the values of vertices, \(A\), \(B\), \(C\) in the Equation (1),

\[\begin{align}\text{Area} &= \frac{1}{2}\left[ {{\text{x}}(2 - 0) + 1(0 - {\text{y}}) + 7({\text{y}} - 2)} \right]\\&0 = \frac{1}{2}\left[ {2{\text{x}} - {\text{y}} + 7{\text{y}} - 14} \right]\\0 &= \frac{1}{2}\left( {2{\text{x}} + 6{\text{y}} - 14} \right)\\2{\text{x}} + 6{\text{y}} - 14 &= 0\\{\text{x}} + 3{\text{y}} - 7 &= 0\end{align}\]

This is the required relation between \(x\) and \(y\).