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

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## Question

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

Video Solution
Coordinate Geometry
Ex 7.4 | Question 2

## 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}(x,y), (1,2)\end{align} and \begin{align}(7,0)\end{align} are collinear.

Steps:

Given,

• \begin{align}\text{Let } A(x_1,y_1) =(x,y)\end{align}
• \begin{align}\text{Let } B(x_2,y_2) =(1,2)\end{align}
• \begin{align}\text{Let } C(x_3,y_3) =(7,0)\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} \begin{bmatrix} x_1 \left( y_2 - y_3 \right) + \\ x_2 \left( y_3 - y_1 \right) + \\ x_3 \left( y_1 - y_2 \right) \end{bmatrix} \;\;\dots(1) \end{align}

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

Area of a triangle

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

This is the required relation between $$x$$ and $$y$$.

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