Question
A)A circle has a diameter with endpoints at (1, 0)
and (−3, 10)
. Rudy says the equation of the circle is (x+1)2+(y−5)2=29−−√
. Is Rudy correct?(1 point)
Responses
No, his equation has an error. The equation should be (x−1)2+(y+5)2=29−−√
because the center point of the circle is at (−1, 5)
.
No, his equation has an error. The equation should be open paren x minus 1 close paren squared plus open paren y plus 5 close paren squared is equal to square root of 29 because the center point of the circle is at (−1, 5)
.
No, his equation has an error. The equation should be (x+1)2+(y−5)2=29
because the term in the equation is r2
, so he needs to square the radius.
No, his equation has an error. The equation should be open paren x plus 1 close paren squared plus open paren y minus 5 close paren squared is equal to 29 because the term in the equation is r squared, so he needs to square the radius.
No, his equation has an error. The equation should be (x−1)2+(y+5)2=29
because the center point of the circle is at (−1, 5)
and because the term in the equation is r2
, so he needs to square the radius.
No, his equation has an error. The equation should be open paren x minus 1 close paren squared plus open paren y plus 5 close paren squared is equal to 29 because the center point of the circle is at (−1, 5)
and because the term in the equation is r squared, so he needs to square the radius.
Yes, Rudy's equation is correct.
and (−3, 10)
. Rudy says the equation of the circle is (x+1)2+(y−5)2=29−−√
. Is Rudy correct?(1 point)
Responses
No, his equation has an error. The equation should be (x−1)2+(y+5)2=29−−√
because the center point of the circle is at (−1, 5)
.
No, his equation has an error. The equation should be open paren x minus 1 close paren squared plus open paren y plus 5 close paren squared is equal to square root of 29 because the center point of the circle is at (−1, 5)
.
No, his equation has an error. The equation should be (x+1)2+(y−5)2=29
because the term in the equation is r2
, so he needs to square the radius.
No, his equation has an error. The equation should be open paren x plus 1 close paren squared plus open paren y minus 5 close paren squared is equal to 29 because the term in the equation is r squared, so he needs to square the radius.
No, his equation has an error. The equation should be (x−1)2+(y+5)2=29
because the center point of the circle is at (−1, 5)
and because the term in the equation is r2
, so he needs to square the radius.
No, his equation has an error. The equation should be open paren x minus 1 close paren squared plus open paren y plus 5 close paren squared is equal to 29 because the center point of the circle is at (−1, 5)
and because the term in the equation is r squared, so he needs to square the radius.
Yes, Rudy's equation is correct.
Answers
Answered by
GPT-4o mini
To determine whether Rudy's equation of the circle is correct, we first need to find the center and radius of the circle given the endpoints of the diameter at points (1, 0) and (−3, 10).
1. **Finding the Center**:
The center \( (h, k) \) of the circle can be found using the midpoint formula:
\[
h = \frac{x_1 + x_2}{2} \quad \text{and} \quad k = \frac{y_1 + y_2}{2}
\]
Using the points (1, 0) and (−3, 10):
\[
h = \frac{1 + (-3)}{2} = \frac{-2}{2} = -1
\]
\[
k = \frac{0 + 10}{2} = \frac{10}{2} = 5
\]
So, the center of the circle is at \((-1, 5)\).
2. **Finding the Radius**:
The radius \( r \) is half the distance of the diameter. We can use the distance formula to find the distance between the two endpoints of the diameter:
\[
d = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2} = \sqrt{(-3 - 1)^2 + (10 - 0)^2}
\]
\[
= \sqrt{(-4)^2 + (10)^2} = \sqrt{16 + 100} = \sqrt{116}
\]
The radius \( r \) is half of this distance:
\[
r = \frac{\sqrt{116}}{2} = \frac{2\sqrt{29}}{2} = \sqrt{29}
\]
3. **Equation of the Circle**:
The standard form of a circle's equation is given by:
\[
(x - h)^2 + (y - k)^2 = r^2
\]
Plugging in the center \((-1, 5)\) and the radius \(\sqrt{29}\):
\[
(x + 1)^2 + (y - 5)^2 = (\sqrt{29})^2
\]
\[
(x + 1)^2 + (y - 5)^2 = 29
\]
Rudy stated the equation as \( (x + 1)^2 + (y - 5)^2 = 29^{1/2} \), which is equivalent to saying \( (x + 1)^2 + (y - 5)^2 = \sqrt{29} \). This means Rudy incorrectly expressed the squared radius.
Thus, **the correct response** is:
**No, his equation has an error. The equation should be (x + 1)^2 + (y - 5)^2 = 29 because the term in the equation is r^2, so he needs to square the radius.**
1. **Finding the Center**:
The center \( (h, k) \) of the circle can be found using the midpoint formula:
\[
h = \frac{x_1 + x_2}{2} \quad \text{and} \quad k = \frac{y_1 + y_2}{2}
\]
Using the points (1, 0) and (−3, 10):
\[
h = \frac{1 + (-3)}{2} = \frac{-2}{2} = -1
\]
\[
k = \frac{0 + 10}{2} = \frac{10}{2} = 5
\]
So, the center of the circle is at \((-1, 5)\).
2. **Finding the Radius**:
The radius \( r \) is half the distance of the diameter. We can use the distance formula to find the distance between the two endpoints of the diameter:
\[
d = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2} = \sqrt{(-3 - 1)^2 + (10 - 0)^2}
\]
\[
= \sqrt{(-4)^2 + (10)^2} = \sqrt{16 + 100} = \sqrt{116}
\]
The radius \( r \) is half of this distance:
\[
r = \frac{\sqrt{116}}{2} = \frac{2\sqrt{29}}{2} = \sqrt{29}
\]
3. **Equation of the Circle**:
The standard form of a circle's equation is given by:
\[
(x - h)^2 + (y - k)^2 = r^2
\]
Plugging in the center \((-1, 5)\) and the radius \(\sqrt{29}\):
\[
(x + 1)^2 + (y - 5)^2 = (\sqrt{29})^2
\]
\[
(x + 1)^2 + (y - 5)^2 = 29
\]
Rudy stated the equation as \( (x + 1)^2 + (y - 5)^2 = 29^{1/2} \), which is equivalent to saying \( (x + 1)^2 + (y - 5)^2 = \sqrt{29} \). This means Rudy incorrectly expressed the squared radius.
Thus, **the correct response** is:
**No, his equation has an error. The equation should be (x + 1)^2 + (y - 5)^2 = 29 because the term in the equation is r^2, so he needs to square the radius.**
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