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Sagot :
To determine whether the point [tex]\((2, -2)\)[/tex] lies on the circle centered at [tex]\((-1, 2)\)[/tex] with a diameter of 10 units, let's go through the necessary steps to verify Amit's claims and calculations.
First, given that the diameter of the circle is 10 units, the radius [tex]\( r \)[/tex] is half of the diameter:
[tex]\[ r = \frac{10}{2} = 5 \text{ units} \][/tex]
Next, we need to calculate the distance between the center of the circle [tex]\((-1, 2)\)[/tex] and the point [tex]\((2, -2)\)[/tex]. The distance formula between two points [tex]\((x_1, y_1)\)[/tex] and [tex]\((x_2, y_2)\)[/tex] is:
[tex]\[ d = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2} \][/tex]
Applying the coordinates of the center [tex]\((-1, 2)\)[/tex] and the point [tex]\((2, -2)\)[/tex]:
[tex]\[ d = \sqrt{(2 - (-1))^2 + (-2 - 2)^2} \][/tex]
[tex]\[ d = \sqrt{(2 + 1)^2 + (-2 - 2)^2} \][/tex]
[tex]\[ d = \sqrt{3^2 + (-4)^2} \][/tex]
[tex]\[ d = \sqrt{9 + 16} \][/tex]
[tex]\[ d = \sqrt{25} \][/tex]
[tex]\[ d = 5 \text{ units} \][/tex]
Since the calculated distance (5 units) is equal to the radius of the circle (5 units), [tex]\((2, -2)\)[/tex] does indeed lie on the circle.
Let's now analyze Amit's work and the statements:
Amit initially tried to calculate the distance:
[tex]\[ \sqrt{(-1 - 2)^2 + (2 - (-2))^2} \][/tex]
But incorrectly calculated the next step:
[tex]\[ \sqrt{(-3)^2 + (4)^2} = \sqrt{9 + 16} = \sqrt{25} = 5 \][/tex]
Thus, the correct calculation leads to a distance of 5 units, which matches the radius. This confirms that Amit's final conclusion that the point [tex]\((2, -2)\)[/tex] is not on the circle is incorrect.
Given the information:
- No, he should have used the origin as the center of the circle. (Incorrect, the center is correctly used as [tex]\((-1, 2)\)[/tex]).
- No, the radius is 10 units, not 5 units. (Incorrect, the radius is 5 units).
- No, he did not calculate the distance correctly. (Correct, as explained, Amit's calculations were incorrect).
- Yes, the distance from the center to [tex]\((2,-2)\)[/tex] is not the same as the radius. (Incorrect, the distance is indeed the same as the radius).
Therefore, the correct statement is:
[tex]\[ \text{No, he did not calculate the distance correctly.} \][/tex]
First, given that the diameter of the circle is 10 units, the radius [tex]\( r \)[/tex] is half of the diameter:
[tex]\[ r = \frac{10}{2} = 5 \text{ units} \][/tex]
Next, we need to calculate the distance between the center of the circle [tex]\((-1, 2)\)[/tex] and the point [tex]\((2, -2)\)[/tex]. The distance formula between two points [tex]\((x_1, y_1)\)[/tex] and [tex]\((x_2, y_2)\)[/tex] is:
[tex]\[ d = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2} \][/tex]
Applying the coordinates of the center [tex]\((-1, 2)\)[/tex] and the point [tex]\((2, -2)\)[/tex]:
[tex]\[ d = \sqrt{(2 - (-1))^2 + (-2 - 2)^2} \][/tex]
[tex]\[ d = \sqrt{(2 + 1)^2 + (-2 - 2)^2} \][/tex]
[tex]\[ d = \sqrt{3^2 + (-4)^2} \][/tex]
[tex]\[ d = \sqrt{9 + 16} \][/tex]
[tex]\[ d = \sqrt{25} \][/tex]
[tex]\[ d = 5 \text{ units} \][/tex]
Since the calculated distance (5 units) is equal to the radius of the circle (5 units), [tex]\((2, -2)\)[/tex] does indeed lie on the circle.
Let's now analyze Amit's work and the statements:
Amit initially tried to calculate the distance:
[tex]\[ \sqrt{(-1 - 2)^2 + (2 - (-2))^2} \][/tex]
But incorrectly calculated the next step:
[tex]\[ \sqrt{(-3)^2 + (4)^2} = \sqrt{9 + 16} = \sqrt{25} = 5 \][/tex]
Thus, the correct calculation leads to a distance of 5 units, which matches the radius. This confirms that Amit's final conclusion that the point [tex]\((2, -2)\)[/tex] is not on the circle is incorrect.
Given the information:
- No, he should have used the origin as the center of the circle. (Incorrect, the center is correctly used as [tex]\((-1, 2)\)[/tex]).
- No, the radius is 10 units, not 5 units. (Incorrect, the radius is 5 units).
- No, he did not calculate the distance correctly. (Correct, as explained, Amit's calculations were incorrect).
- Yes, the distance from the center to [tex]\((2,-2)\)[/tex] is not the same as the radius. (Incorrect, the distance is indeed the same as the radius).
Therefore, the correct statement is:
[tex]\[ \text{No, he did not calculate the distance correctly.} \][/tex]
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