A circle, has its centre at (-1, 3). If one end of a diameter of the circle has co-ordinates (2,5), then find the co-ordinates of the other end of the diameter.

Options

A. (-4,1)
B. (1,8)
C. (0.5, 4)
D. (-1,4)

Answer

To find the coordinates of the other end of the diameter of the circle, we can use the concept of the midpoint of a line segment. The center of the circle is the midpoint of the diameter.

Let's denote:
- The center of the circle (midpoint) as \( C(-1, 3) \)
- One end of the diameter as \( A(2, 5) \)
- The other end of the diameter as \( B(x, y) \)

### Step 1: Use the midpoint formula
The midpoint \( C \) of a line segment with endpoints \( A(x_1, y_1) \) and \( B(x_2, y_2) \) is given by:
\[ C = \left( \frac{x_1 + x_2}{2}, \frac{y_1 + y_2}{2} \right) \]
In our case, we have:
\[ C = (-1, 3), \quad A = (2, 5), \quad B = (x, y) \]

### Step 2: Set up the equations
From the midpoint formula, we can set up the following equations:
\[ -1 = \frac{2 + x}{2} \quad \text{(1)} \]
\[ 3 = \frac{5 + y}{2} \quad \text{(2)} \]

### Step 3: Solve for \( x \)
From equation (1):
\[ -1 = \frac{2 + x}{2} \]
Multiply both sides by 2:
\[ -2 = 2 + x \]
Now, isolate \( x \):
\[ x = -2 - 2 = -4 \]

### Step 4: Solve for \( y \)
From equation (2):
\[ 3 = \frac{5 + y}{2} \]
Multiply both sides by 2:
\[ 6 = 5 + y \]
Now, isolate \( y \):
\[ y = 6 - 5 = 1 \]

### Step 5: Write the final coordinates
Thus, the coordinates of the other end of the diameter \( B \) are:
\[ B(-4, 1) \]

### Final Answer:
The coordinates of the other end of the diameter are \( (-4, 1) \).

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