The circle $$ C $$ has equation $$ x^{2}+y^{2}-10 x+6 y+30=0 $$ Find the coordinates of the centre of $$ c $$.

Question

UpStudy AI · Accepted Answer

To find the coordinates of the center of the circle given by the equation

$$
x^{2}+y^{2}-10 x+6 y+30=0,
$$

we can rewrite the equation in standard form. The standard form of a circle's equation is

$$
(x - h)^2 + (y - k)^2 = r^2,
$$

where $$(h, k)$$ is the center of the circle and $$r$$ is the radius.

### Step 1: Rearranging the equation

We start by rearranging the given equation:

$$
x^{2} - 10x + y^{2} + 6y + 30 = 0.
$$

### Step 2: Completing the square

Next, we will complete the square for the $$x$$ and $$y$$ terms.

**For the $$x$$ terms:**

1. Take the coefficient of $$x$$, which is $$-10$$, halve it to get $$-5$$, and square it to get $$25$$.
2. Rewrite the $$x$$ terms:

$$
x^{2} - 10x = (x - 5)^{2} - 25.
$$

**For the $$y$$ terms:**

1. Take the coefficient of $$y$$, which is $$6$$, halve it to get $$3$$, and square it to get $$9$$.
2. Rewrite the $$y$$ terms:

$$
y^{2} + 6y = (y + 3)^{2} - 9.
$$

### Step 3: Substitute back into the equation

Now substitute these completed squares back into the equation:

$$
(x - 5)^{2} - 25 + (y + 3)^{2} - 9 + 30 = 0.
$$

Combine the constants:

$$
(x - 5)^{2} + (y + 3)^{2} - 4 = 0.
$$

This simplifies to:

$$
(x - 5)^{2} + (y + 3)^{2} = 4.
$$

### Step 4: Identify the center

From the standard form $$(x - h)^{2} + (y - k)^{2} = r^{2}$$, we can identify the center $$(h, k)$$:

- $$h = 5$$
- $$k = -3$$

### Conclusion

Thus, the coordinates of the center of the circle $$C$$ are

$$
\boxed{(5, -3)}.
$$
The center of the circle is at the point (5, -3).

The circle \( C \) has equation \[ x^{2}+y^{2}-10 x+6 y+30=0 \] Find the coordinates of the centre of \( c \).