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Activity 1: Write the following equations of a circle to their general forms.

1. (x + 4)² + (y - 7)² = 100

2. (x - 1)² + (y - 4)² = 64

3. (x - 2)² + (y - 1)² = 11²

4. (x + 1)² + (y + 2)² = 25

5. (x - 2)² + (y - 4)² = 72 ​ ​ ​

Sagot :

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Answer:

1.   x² + y² + 8x - 14y - 35 = 0

2.  x² + y² - 2x - 8y - 47 = 0

3.  x² + y² - 4x - 2y - 116 = 0

4.  x² + y² + 2x + 4y - 20 = 0

5.  x² + y² - 4x - 8y - 52 = 0

Step-by-step explanation:

General equation of a circle is:  x² + y² + 2gx + 2fy + c = 0

with center (-g, -f) and radius √(g² + f² - c)

1.  From inspection:  center (-4, 7)  and radius = √100 = 10

   Therefore, g = 4  and  f = -7

   √(4² + (-7)² - c) = 10  ⇒  65 - c = 100  ⇒  c = -35

   So,  x² + y² + 8x - 14y - 35 = 0

2.  From inspection:  center (1, 4)  and radius = √64 = 8

   Therefore, g = -1  and  f = -4

   √((-1)² + (-4)² - c) = 8  ⇒  17 - c = 64  ⇒  c = -47

   So,  x² + y² - 2x - 8y - 47 = 0

3.  From inspection:  center (2, 1)  and radius = √(11²) = 11

   Therefore, g = -2  and  f = -1

   √((-2)² + (-1)² - c) = 11  ⇒  5 - c = 121  ⇒  c = -116

   So,  x² + y² - 4x - 2y - 116 = 0

4.  From inspection:  center (-1, -2)  and radius = √25 = 5

   Therefore, g = 1  and  f = 2

   √(1² + 2² - c) = 5  ⇒  5 - c = 25  ⇒  c = -20

   So,  x² + y² + 2x + 4y - 20 = 0

5.  From inspection:  center (2, 4)  and radius = √72

   Therefore, g = -2  and  f = -4

   √((-2)² + (-4)² - c) = √72  ⇒  20 - c = 72  ⇒  c = -52

   So,  x² + y² - 4x - 8y - 52 = 0

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