Question

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

Answer 1

Finding the coordinates of $A$:

It is known that in a circle, the center divides the diameter into two equal parts, that is, the center is the mid-point of the diameter.

Also,

It is known that the coordinates of the mid-point of a line, whose endpoints are $\left(x_1,y_1\right)\mathrm{and}\left(x_2,y_2\right)$ can be found by using the formula given below:

$\left(x,y\right)=\left(\frac{x_1+x_2}{2},\frac{y_1+y_2}{2}\right)$ …(1)

Here,

$\left(x,y\right)=\left(2,-3\right)\mathrm{and}\left(x_2,y_2\right)=\left(1,4\right)$

Substitute the known values in equation (1):

$$\begin{gathered} \therefore \left(2,-3\right)=\left(\frac{x_1+1}{2},\frac{y_1+4}{2}\right) \\ \Rightarrow \frac{x_1+1}{2}=2\mathrm{and}\frac{y_1+4}{2}=-3 \\ \Rightarrow x_1+1=4\mathrm{and}{y}_1+4=-6 \\ \Rightarrow x_1=3\mathrm{and}y=-10 \end{gathered}$$

Final Answer:

Hence, the coordinates of point $A$ is $\left(3,-10\right)$.