Find the coordinates of a point A, where AB is the diameter of a circle whose centre is and B is .

Question

Homi · Accepted Answer

AB is a diameter, so the centre of the circle lies exactly at the midpoint of AB. We know the centre and one endpoint B — so we use the midpoint formula in reverse to find A.

Midpoint Formula: The midpoint of a line segment joining $\text{A}(x_1, y_1)$ and $\text{B}(x_2, y_2)$ :

$M = \left(\frac{x_1 + x_2}{2},\ \frac{y_1 + y_2}{2}\right)$

The centre of a circle is the midpoint of its diameter.

Step 1 — Identify given points

Let's write down what we know. The centre of the circle is $\mathbf{C(2, -3)}$ . Point B is $\mathbf{B(1, 4)}$ . Let point A be $\mathbf{A(x, y)}$ . AB is the diameter of the circle.

Diagram 1

Step 2 — Use the midpoint formula

We know the midpoint formula. The coordinates of the midpoint $(x_m, y_m)$ are $((x_1 + x_2)/2, (y_1 + y_2)/2)$ . Here, C is the midpoint of AB. So, we can set up equations for x and y coordinates.

Let's find the x-coordinate of A. $2 = \frac{x + 1}{2}$ $2 \times 2 = x + 1$ $4 = x + 1$ $x = 4 - 1$

$\boxed{x = 3}$

Now, let's find the y-coordinate of A. $-3 = \frac{y + 4}{2}$ $-3 \times 2 = y + 4$ $-6 = y + 4$ $y = -6 - 4$

$\boxed{y = -10}$

Answer

The coordinates of point A are $\mathbf{(3, -10)}$ .

Question 7

Answer

More questions in Exercise 7.2