The co-ordinates of one end-point of a circle are (-3, 1) and the co-ordinates of the centre of the circle are (2, 4). The co-ordinates of the other end-point of the diameter are

Options

A. `((-1)/(2), (-3)/(2))`
B. `(-7, 9)`
C. `(7, 9)`
D. `((1)/(2), (3)/(2))`

Answer

To find the coordinates of the other end-point of the diameter of the circle, we can use the properties of the midpoint. The center of the circle is the midpoint of the diameter.

Let's denote the coordinates of the given end-point as \( A(-3, 1) \) and the center of the circle as \( C(2, 4) \). We need to find the coordinates of the other end-point \( B(x, y) \).

### 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 the formula:
\[ C\left(\frac{x_1 + x_2}{2}, \frac{y_1 + y_2}{2}\right) \]
In our case, the coordinates of \( A \) are \( (-3, 1) \) and the coordinates of \( C \) are \( (2, 4) \).

### Step 2: Set up equations for the x-coordinates
From the midpoint formula, we can set up the equation for the x-coordinates:
\[ 2 = \frac{-3 + x}{2} \]
To eliminate the fraction, multiply both sides by 2:
\[ 4 = -3 + x \]
Now, solve for \( x \):
\[ x = 4 + 3 = 7 \]

### Step 3: Set up equations for the y-coordinates
Next, we set up the equation for the y-coordinates:
\[ 4 = \frac{1 + y}{2} \]
Again, multiply both sides by 2:
\[ 8 = 1 + y \]
Now, solve for \( y \):
\[ y = 8 - 1 = 7 \]

### Step 4: Conclusion
Thus, the coordinates of the other end-point \( B \) are \( (7, 7) \).

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

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