One end of the diameter of a circle is (-2, 5). Find the co-ordinates of the other end of it, if the centre of the circle is (2, -1).

Answer

To find the coordinates of the other end of the diameter of the circle, we can use the section formula. Here’s a step-by-step solution:

### Step 1: Identify the given points
We have:
- One end of the diameter (Point A) = (-2, 5)
- Center of the circle (Point O) = (2, -1)

Let the other end of the diameter (Point B) be (P, Q).

### Step 2: Understand the relationship
Since the center of the circle divides the diameter into two equal halves, the ratio of the segments AO and OB is 1:1.

### Step 3: Apply the section formula
The section formula states that if a point divides a line segment joining two points (x1, y1) and (x2, y2) in the ratio m:n, then the coordinates of the dividing point (x, y) are given by:
\[ x = \frac{m \cdot x_2 + n \cdot x_1}{m + n} \]
\[ y = \frac{m \cdot y_2 + n \cdot y_1}{m + n} \]

Here, m = 1, n = 1, (x1, y1) = (-2, 5), and (x2, y2) = (P, Q).

### Step 4: Set up the equations for x and y coordinates
Using the section formula for the x-coordinate:
\[ 2 = \frac{1 \cdot P + 1 \cdot (-2)}{1 + 1} \]
This simplifies to:
\[ 2 = \frac{P - 2}{2} \]

Using the section formula for the y-coordinate:
\[ -1 = \frac{1 \cdot Q + 1 \cdot 5}{1 + 1} \]
This simplifies to:
\[ -1 = \frac{Q + 5}{2} \]

### Step 5: Solve for P
From the x-coordinate equation:
\[ 2 \cdot 2 = P - 2 \]
\[ 4 = P - 2 \]
\[ P = 4 + 2 = 6 \]

### Step 6: Solve for Q
From the y-coordinate equation:
\[ -1 \cdot 2 = Q + 5 \]
\[ -2 = Q + 5 \]
\[ Q = -2 - 5 = -7 \]

### Step 7: Conclusion
Thus, the coordinates of the other end of the diameter (Point B) are:
\[ (P, Q) = (6, -7) \]

### Final Answer
The coordinates of the other end of the diameter are (6, -7).

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