# 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 $$(3, -10)$$

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