Find the best answers to your questions with the help of IDNLearn.com's knowledgeable users. Whether it's a simple query or a complex problem, our community has the answers you need.
Sagot :
To determine which equation represents a circle with a center at [tex]\( Z(-3, 5) \)[/tex] and a radius of 4 units, we need to use the standard form of the equation of a circle:
[tex]\[ (x - h)^2 + (y - k)^2 = r^2 \][/tex]
where [tex]\((h, k)\)[/tex] is the center of the circle and [tex]\(r\)[/tex] is the radius.
### Step-by-Step Solution:
1. Identify the center [tex]\((h, k)\)[/tex] and the radius [tex]\(r\)[/tex]:
- The center [tex]\((h, k)\)[/tex] is given as [tex]\((-3, 5)\)[/tex].
- The radius [tex]\(r\)[/tex] is given as 4 units.
2. Substitute the values into the standard equation:
- [tex]\(h = -3\)[/tex]
- [tex]\(k = 5\)[/tex]
- [tex]\(r = 4\)[/tex]
3. Write the equation with the given center and radius:
- Substitute [tex]\(h = -3\)[/tex] and [tex]\(k = 5\)[/tex] into the equation:
[tex]\[ (x - (-3))^2 + (y - 5)^2 = 4^2 \][/tex]
- Simplify the terms:
[tex]\[ (x + 3)^2 + (y - 5)^2 = 16 \][/tex]
4. Match the derived equation with the given options:
Let's look at each of the options provided:
A. [tex]\((x-3)^2 + (y+5)^2 = 4\)[/tex]
- Here, the center would be [tex]\((3, -5)\)[/tex] and the radius squared would be 4, which does not match our parameters.
B. [tex]\((x-3)^2 + (y+5)^2 = 16\)[/tex]
- Here, the center would be [tex]\((3, -5)\)[/tex] and the radius squared is 16. Though the radius squared is correct, the center does not match our parameters.
C. [tex]\((x+3)^2 + (y-5)^2 = 4\)[/tex]
- Here, the center would be [tex]\((-3, 5)\)[/tex] but the radius squared is 4, which does not match our parameters.
D. [tex]\((x+3)^2 + (y-5)^2 = 16\)[/tex]
- Here, the center is [tex]\((-3, 5)\)[/tex] and the radius squared is 16. This perfectly matches our derived equation.
The correct answer is:
[tex]\[ \boxed{(x+3)^2 + (y-5)^2 = 16} \][/tex]
Thus, the equation representing a circle with center [tex]\( Z(-3, 5) \)[/tex] and a radius of 4 units is:
D. [tex]\((x+3)^2+(y-5)^2=16\)[/tex]
[tex]\[ (x - h)^2 + (y - k)^2 = r^2 \][/tex]
where [tex]\((h, k)\)[/tex] is the center of the circle and [tex]\(r\)[/tex] is the radius.
### Step-by-Step Solution:
1. Identify the center [tex]\((h, k)\)[/tex] and the radius [tex]\(r\)[/tex]:
- The center [tex]\((h, k)\)[/tex] is given as [tex]\((-3, 5)\)[/tex].
- The radius [tex]\(r\)[/tex] is given as 4 units.
2. Substitute the values into the standard equation:
- [tex]\(h = -3\)[/tex]
- [tex]\(k = 5\)[/tex]
- [tex]\(r = 4\)[/tex]
3. Write the equation with the given center and radius:
- Substitute [tex]\(h = -3\)[/tex] and [tex]\(k = 5\)[/tex] into the equation:
[tex]\[ (x - (-3))^2 + (y - 5)^2 = 4^2 \][/tex]
- Simplify the terms:
[tex]\[ (x + 3)^2 + (y - 5)^2 = 16 \][/tex]
4. Match the derived equation with the given options:
Let's look at each of the options provided:
A. [tex]\((x-3)^2 + (y+5)^2 = 4\)[/tex]
- Here, the center would be [tex]\((3, -5)\)[/tex] and the radius squared would be 4, which does not match our parameters.
B. [tex]\((x-3)^2 + (y+5)^2 = 16\)[/tex]
- Here, the center would be [tex]\((3, -5)\)[/tex] and the radius squared is 16. Though the radius squared is correct, the center does not match our parameters.
C. [tex]\((x+3)^2 + (y-5)^2 = 4\)[/tex]
- Here, the center would be [tex]\((-3, 5)\)[/tex] but the radius squared is 4, which does not match our parameters.
D. [tex]\((x+3)^2 + (y-5)^2 = 16\)[/tex]
- Here, the center is [tex]\((-3, 5)\)[/tex] and the radius squared is 16. This perfectly matches our derived equation.
The correct answer is:
[tex]\[ \boxed{(x+3)^2 + (y-5)^2 = 16} \][/tex]
Thus, the equation representing a circle with center [tex]\( Z(-3, 5) \)[/tex] and a radius of 4 units is:
D. [tex]\((x+3)^2+(y-5)^2=16\)[/tex]
Your participation means a lot to us. Keep sharing information and solutions. This community grows thanks to the amazing contributions from members like you. Your search for solutions ends here at IDNLearn.com. Thank you for visiting, and come back soon for more helpful information.