question_answer

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

A)  \[4,-\,\,5\]

B)                     \[-\,\,7,11\]

C)   3, 9                            

D)  \[3,-\,\,10\]

Correct Answer: D

Solution :

Suppose, AB be a diameter of the circle having its centre at C \[(2,-\,\,3)\]and coordinates of end point B are (1, 4).

Let the coordinates of A be (x, y)

Since, AB is diameter.

\[\therefore \]C is the mid-point of AB

The coordinates of C are\[\left( \frac{x+1}{2},\frac{y+4}{2} \right)\]

[\[\because \]Coordinate of mid \[-\] point

\[\left. =\left( \frac{{{x}_{1}}+{{x}_{2}}}{2},\frac{{{y}_{1}}+{{y}_{2}}}{2} \right) \right]\]

But it is given that the coordinates of C are \[(2,-\,\,3).\]

\[\therefore \]      \[\frac{x+1}{2}=2\]\[\Rightarrow \]\[x+1=4\]

\[\Rightarrow \]   \[x=3\]and \[\frac{y+4}{2}=-\,\,3\]

\[\Rightarrow \]   \[y+4=-\,\,6\]

\[\Rightarrow \]   \[y=-10\]

So the required coordinates of A are \[(3,-\,\,10).\]