Let's analyze the problem step by step to determine if Amit's work is correct.
1. Identify the Center and Radius of the Circle:
- The center of the circle is given as [tex]\((-1, 2)\)[/tex].
- The diameter of the circle is 10 units, so the radius [tex]\( r \)[/tex] is half of the diameter:
[tex]\[
r = \frac{10}{2} = 5 \text{ units}
\][/tex]
2. Find the Distance from the Center to the Point [tex]\((2, -2)\)[/tex]:
- The coordinates of the given point are [tex]\((2, -2)\)[/tex].
- To find the distance between the center [tex]\((-1, 2)\)[/tex] and the point [tex]\((2, -2)\)[/tex], we use the distance formula:
[tex]\[
\text{Distance} = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2}
\][/tex]
where [tex]\((x_1, y_1)\)[/tex] are the coordinates of the center and [tex]\((x_2, y_2)\)[/tex] are the coordinates of the point.
- Substituting the values:
[tex]\[
\text{Distance} = \sqrt{(2 - (-1))^2 + (-2 - 2)^2}
\][/tex]
Simplifying inside the square root:
[tex]\[
\text{Distance} = \sqrt{(2 + 1)^2 + (-2 - 2)^2} = \sqrt{3^2 + (-4)^2} = \sqrt{9 + 16} = \sqrt{25} = 5
\][/tex]
3. Compare the Distance to the Radius:
- The calculated distance from the center to the point [tex]\((2, -2)\)[/tex] is 5 units.
- The radius of the circle is also 5 units.
4. Determine if the Point Lies on the Circle:
- For a point to lie on the circle, the distance from the center to the point must be equal to the radius.
- Since the distance from the center to [tex]\((2, -2)\)[/tex] is 5 units, which is equal to the radius, [tex]\((2, -2)\)[/tex] lies on the circle.
5. Verify Amit's Work:
- Amit stated that the point [tex]\((2, -2)\)[/tex] does not lie on the circle because he calculated the distance incorrectly as 3 units. His steps and conclusion appear flawed.
Hence, Amit's work is incorrect. The correct distance calculation shows that the point [tex]\((2, -2)\)[/tex] lies on the circle. The correct findings are:
- The radius is 5 units.
- The distance from the center [tex]\((-1, 2)\)[/tex] to the point [tex]\((2, -2)\)[/tex] is 5 units.
- Therefore, the point [tex]\((2, -2)\)[/tex] does lie on the circle.
