Write the standard form of the equation and the general form of the equation of the circle with radius $$ r $$ and center $$ (h, k) $$. Then graph the circle. $$ r=5 ; \quad(h, k)=(-4,-3) $$ The standard form of the equation of this circle is $$ (x+4)^{2}+(y+3)^{2}=25 $$. The general form of the equation of this circle is (Simplify your answer.)

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We start with the standard form of a circle with center (h, k) and radius r:

(x – h)² + (y – k)² = r²

Given h = –4, k = –3, and r = 5, the standard form becomes

(x + 4)² + (y + 3)² = 25

To express this in the general form, we expand and simplify the equation:

1. Expand (x + 4)² and (y + 3)²:
  (x + 4)² = x² + 8x + 16
  (y + 3)² = y² + 6y + 9

2. Combine the expansions:
  x² + 8x + 16 + y² + 6y + 9 = 25

3. Combine like terms (16 + 9 = 25):
  x² + y² + 8x + 6y + 25 = 25

4. Subtract 25 from both sides to write in general form:
  x² + y² + 8x + 6y = 0

So, the general form of the equation is:

x² + y² + 8x + 6y = 0

Graphing the Circle:

• Center: (–4, –3)
• Radius: 5

Key points on the circle can be found by moving 5 units in the cardinal directions from the center:
  – Rightmost point: (–4 + 5, –3) = (1, –3)
  – Leftmost point: (–4 – 5, –3) = (–9, –3)
  – Topmost point: (–4, –3 + 5) = (–4, 2)
  – Bottommost point: (–4, –3 – 5) = (–4, –8)

To sketch the graph:
 1. Plot the center at (–4, –3).
 2. Plot the four key points listed above.
 3. Draw a smooth curve passing through these points to form the circle.

This completes the answer:

Standard form: (x + 4)² + (y + 3)² = 25  
General form: x² + y² + 8x + 6y = 0
The general form of the equation is $$ x^{2} + y^{2} + 8x + 6y = 0 $$.

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