Centre and radius of the circle $2x^{2}+2y^{2}+16x+20y-16=0$ be

Question

Accepted Answer

To determine the center and radius of the circle given by the equation $2x^{2} + 2y^{2} + 16x + 20y - 16 = 0$, we first need to rewrite the equation in the standard form of a circle, $(x - h)^2 + (y - k)^2 = r^2$, where $(h, k)$ is the center and $r$ is the radius.

### Step-by-step Explanation:

1. **Simplify the Equation:**
   Divide the entire equation by 2 to simplify:
   $$
   x^2 + y^2 + 8x + 10y - 8 = 0
   $$

2. **Complete the Square:**
   - For the $x$ terms: $x^2 + 8x$
     - Take half of 8, which is 4, and square it to get 16.
     - Add and subtract 16 inside the equation.
   - For the $y$ terms: $y^2 + 10y$
     - Take half of 10, which is 5, and square it to get 25.
     - Add and subtract 25 inside the equation.

The equation becomes:
   $$
   (x^2 + 8x + 16) + (y^2 + 10y + 25) = 8 + 16 + 25
   $$

3. **Rewrite as Perfect Squares:**
   $$
   (x + 4)^2 + (y + 5)^2 = 49
   $$

Here, $(x + 4)^2$ and $(y + 5)^2$ are perfect squares.

4. **Identify the Center and Radius:**
   - The center $(h, k)$ is $(-4, -5)$.
   - The radius $r$ is $\sqrt{49} = 7$.

### Conclusion:

- **Option 1**: $(-4, -5); 7$ is correct because the center of the circle is $(-4, -5)$ and the radius is 7, as derived from the equation.

### Explanation of Incorrect Options:

- **Option 2**: $(-8, -10); 14$ is incorrect because the center and radius do not match the derived values from the equation.
- **Option 3**: $(4, 5); 7$ is incorrect because the center is not $(4, 5)$.
- **Option 4**: $(8, 10); 14$ is incorrect because neither the center nor the radius matches the derived values.

Thus, the correct answer is indeed Option 1, as it accurately reflects the center and radius of the circle derived from the given equation.