question_answer

The coordinates of point A, where AB is the diameter of a circle whose centre is (3, -4) and B is (1,4) is

A) (2, 0)

B) (12, -5)

C) (5, -12)

D) None of the above

Correct Answer: C

Solution :

Let, AB be the diameter and C be the centre of the circle. Let coordinates of A be (a, b). Clearly, C will be the mid-point of AB.

\[\therefore\]Coordinates of \[C=\left( \frac{a+1}{2},\,\frac{b+4}{2} \right)\]

\[\left[ \because \,\,mid\,-\,po\operatorname{int}=\left( \frac{{{x}_{1}}+{{x}_{2}}}{2},\frac{{{y}_{1}}+{{y}_{2}}}{2} \right) \right]\]

\[\Rightarrow \,\,\,\left( 3,\,\,-4 \right)=\left( \frac{a+1}{2},\frac{b+4}{2} \right)\]

[given, coordinates of C = (3, - 4)] On comparing the coordinates of x and y from both sides, we get

\[\frac{a+1}{2}=3\]  and \[\frac{b+4}{2}=-4\]

\[\Rightarrow a+1=6\] and          \[b+4=-8\]

\[\Rightarrow \,\,\,a=5\] and \[b=-12\]

Hence, the coordinates of point

A are (5, -12).