Ex.7.2 Q7 Coordinate Geometry Solution - NCERT Maths Class 10

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Question

Find the coordinates of a point \(A\), where \(AB\) is the diameter of circle whose center is \((2, -3)\) and \(B\) is \((1, 4)\)

 

Text Solution

 

Reasoning:

The coordinates of the point \(P(x, y)\) which divides the line segment joining the points \(A(x1, y1)\) and \(B(x2, y2)\), internally, in the ratio \(\rm m1 : m2\) is given by the Section Formula.

What is the known?

The \(x\) and \(y\) co-ordinates of the center of the circle and one end of the diameter \(B\).

What is the unknown?

The coordinates of a point \(A\).

Steps:

From the Figure,

Given,

  • Let the coordinates of point \(A\) be \((x, y)\).
  • Mid-point of \(AB\) is \(C\) \((2, -3)\), which is the center of the circle.

\(\begin{align} \therefore \; (2, - 3) &= \left( {\frac{{{\text{x}} + 1}}{2},\frac{{{\text{y}} + 4}}{2}} \right)   \end{align}\)

\(\begin{align}  \Rightarrow \frac{{{\text{x}} + 1}}{2}& = 2{\text{ and }}\frac{{{\text{y}} + 4}}{2} = - 3 & &  \end{align}\) (By Cross multiplying & transposing)

\(\begin{align}  \Rightarrow {\text{x}} + 1 &= 4{\text{ and y}} + 4 = - 6  \end{align}\)

\(\begin{align}  \Rightarrow {\text{x}} &= 3{\text{ and y}} = - 10\end{align}\) 

Therefore, the coordinates of \(A\) are \((3, -10)\)