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 follow these steps:

### Step 1: Identify the known points
We are given:
- The center of the circle \( O \) at coordinates \( (-1, 3) \).
- One end of the diameter \( P \) at coordinates \( (2, 5) \).

### Step 2: Use the midpoint formula
The center of the circle is the midpoint of the diameter. If we denote the other end of the diameter as \( Q(a, b) \), we can use the midpoint formula:
\[ O = \left( \frac{x_1 + x_2}{2}, \frac{y_1 + y_2}{2} \right) \]
where \( (x_1, y_1) \) are the coordinates of point \( P \) and \( (x_2, y_2) \) are the coordinates of point \( Q \).

### Step 3: Set up the equations
From the midpoint formula, we have:
\[ \left( \frac{2 + a}{2}, \frac{5 + b}{2} \right) = (-1, 3) \]
This gives us two equations:
1. \( \frac{2 + a}{2} = -1 \)
2. \( \frac{5 + b}{2} = 3 \)

### Step 4: Solve the first equation for \( a \)
Starting with the first equation:
\[ \frac{2 + a}{2} = -1 \]
Multiply both sides by 2:
\[ 2 + a = -2 \]
Now, subtract 2 from both sides:
\[ a = -2 - 2 = -4 \]

### Step 5: Solve the second equation for \( b \)
Now, solve the second equation:
\[ \frac{5 + b}{2} = 3 \]
Multiply both sides by 2:
\[ 5 + b = 6 \]
Subtract 5 from both sides:
\[ b = 6 - 5 = 1 \]

### Step 6: Write the coordinates of point \( Q \)
Now that we have both \( a \) and \( b \), we can write the coordinates of point \( Q \):
\[ Q = (a, b) = (-4, 1) \]

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

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