do we have to inverse the sign of h and k when using them fir centre directly.from equation

Yes, you do need to change the signs of \( h \) and \( k \) when identifying the center of the circle directly from the equation.

### Explanation

The standard form of the equation of a circle is:

\[
(x - h)^2 + (y - k)^2 = r^2
\]

Here, \( h \) and \( k \) are the coordinates of the center of the circle \((h, k)\).

- If the equation is written as \((x - h)^2\), then the \( h \) value in the center is positive.
- If the equation is written as \((x + h)^2\), it means \( h \) is negative (because \( x + h \) is equivalent to \( x - (-h) \)).

Similarly, for the \( y \) term:

- If the equation is written as \((y - k)^2\), then \( k \) is positive.
- If the equation is written as \((y + k)^2\), it means \( k \) is negative.

### Example

Consider the equation:

\[
(x - 3)^2 + (y + 4)^2 = 25
\]

- For the \( x \)-term \((x - 3)\), \( h = 3 \).
- For the \( y \)-term \((y + 4)\), \( k = -4 \).

So, the center of the circle is \((3, -4)\).

### Why Do We Inverse the Signs?

In the standard form \((x - h)\) and \((y - k)\), \( h \) and \( k \) are the coordinates of the center. However, in the equation, they're written as subtracted values. This means:

- When you see \((x - h)\), the \( h \) from the equation is directly the coordinate for the center.
- When you see \((x + h)\), it's actually \((x - (-h))\), meaning the coordinate \( h \) is negative.

So, you must inverse the sign to correctly identify the center.

Would you like more details or have any questions?

Here are 8 questions to deepen your understanding:
1. How can you find the center of a circle if the equation is given in expanded form?
2. How do you complete the square when given a general equation of a circle?
3. Can you derive the equation of a circle from its geometric definition?
4. How does shifting a circle horizontally or vertically affect its equation?
5. What is the significance of the sign change in the standard form of the circle equation?
6. How do you find the distance between two circles using their equations?
7. Can the equation of a circle be represented differently depending on the given information?
8. What happens to the circle's equation if it is reflected over the x-axis or y-axis?

### Tip:
Always double-check the signs for the center coordinates when interpreting a circle's equation—it's a common area for mistakes on exams!

Explore how to correctly identify the center of a circle from its equation. Learn about the standard form of the circle equation, where coordinates h and k must have their signs inverted for accurate center calculation. Suitable for students in Grades 9-12.

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