question_answer

  The coordinates of a point A, where AB is diameter of a circle whose centre is \[(2,-3)\] and B is the point \[(1,4)\] are: (CBSE 2019)

A) \[(3,-10)\]

B) \[(-3,10)\]

C) \[(3,10)\]

D) \[(-3,-10)\]

Correct Answer: A

Solution :

[a] Let the coordinate of point A be \[({{x}_{1}},{{y}_{1}}).\]

Centre C of a circle is the mid-point of AB.

\[\therefore \] Mid-point of \[AB=\left( \frac{{{x}_{1}}+1}{2},\frac{{{y}_{1}}+4}{2} \right).\]

\[\Rightarrow \,\,\,\,\,\,\,\,\,\,\,\,\,\,(2,-3)=\left( \frac{{{x}_{1}}+1}{2},\frac{{{y}_{1}}+4}{2} \right)\]

On equating the x and y coordinates, we get

3+\[2=\frac{{{x}_{1}}+1}{2}\] and \[-3=\frac{{{y}_{1}}+4}{2}\]

\[\Rightarrow \,\,\,\,\,\,\,{{x}_{1}}=4-1\] and \[{{y}_{1}}=-6-4\]

\[\Rightarrow \,\,\,\,\,\,\,{{x}_{1}}=3\] and \[{{y}_{1}}=-10\]

Hence, coordinate of point A is \[(3,-10).\]