Conic Sections and Circle

1. Given the center at (0,0) and the radius is 5, find the equation of the circle.

X2 + y2 =10

X2 + y2 =25

X2 - y2 =10

X2 - y2 =25

The equation of a circle with center (0,0) and radius 5 is x^2 + y^2 = 25. This is because the equation of a circle with center (h,k) and radius r is (x-h)^2 + (y-k)^2 = r^2. In this case, h=0, k=0, and r=5, so plugging in these values gives us x^2 + y^2 = 25.

Explanation

The equation of a circle with center (0,0) and radius 5 is x^2 + y^2 = 25. This is because the equation of a circle with center (h,k) and radius r is (x-h)^2 + (y-k)^2 = r^2. In this case, h=0, k=0, and r=5, so plugging in these values gives us x^2 + y^2 = 25.

2. What is the standard form of the equation x²+y²+14x-12y+4=0

Option 1

Option 2

Option 3

Option 4

The standard form of the equation x^2 + y^2 + 14x - 12y + 4 = 0 is obtained by rearranging the terms to have the x term and y term separately on one side of the equation and the constant term on the other side. This can be done by completing the square for both the x and y terms. The equation can then be written as (x + 7)^2 - 49 + (y - 6)^2 - 36 + 4 = 0, which simplifies to (x + 7)^2 + (y - 6)^2 = 81. Therefore, the correct answer is Option 3.

Explanation

The standard form of the equation x^2 + y^2 + 14x - 12y + 4 = 0 is obtained by rearranging the terms to have the x term and y term separately on one side of the equation and the constant term on the other side. This can be done by completing the square for both the x and y terms. The equation can then be written as (x + 7)^2 - 49 + (y - 6)^2 - 36 + 4 = 0, which simplifies to (x + 7)^2 + (y - 6)^2 = 81. Therefore, the correct answer is Option 3.

3. Which of the following is the center and the radius of this equation

Center (-3,13) radius= 4

Center (3,13) radius= 4

Center (-3,-13) radius= 4

Center (3,-13) radius= 4

The correct answer is center (3,13) & radius= 4. This is because the center of a circle is represented by the coordinates (h, k), where h is the x-coordinate and k is the y-coordinate. In this case, the center is (3,13), indicating that the circle is centered at the point (3,13) on the coordinate plane. The radius of a circle is the distance from the center to any point on the circle. Given that the radius is 4, it means that the distance from the center to any point on the circle is 4 units.

Explanation

The correct answer is center (3,13) & radius= 4. This is because the center of a circle is represented by the coordinates (h, k), where h is the x-coordinate and k is the y-coordinate. In this case, the center is (3,13), indicating that the circle is centered at the point (3,13) on the coordinate plane. The radius of a circle is the distance from the center to any point on the circle. Given that the radius is 4, it means that the distance from the center to any point on the circle is 4 units.

4. This is a set of all coplanar points such that the distance from a fixed point is constant.

Circle

Parabola

Hyperbola

Ellipse

A circle is defined as a set of all coplanar points that are equidistant from a fixed point called the center. The distance from any point on the circle to the center remains constant, which is the radius of the circle. Therefore, the given description perfectly matches the characteristics of a circle.

Explanation

A circle is defined as a set of all coplanar points that are equidistant from a fixed point called the center. The distance from any point on the circle to the center remains constant, which is the radius of the circle. Therefore, the given description perfectly matches the characteristics of a circle.

5. What is the equation of the circle whose Center is (−6, −15) and radius: square root of 5

Option 1

Option 2

Option 3

Option 4

Option 5

The equation of a circle with center (h, k) and radius r is (x - h)^2 + (y - k)^2 = r^2. In this case, the center is (-6, -15) and the radius is the square root of 5. Plugging these values into the equation, we get (x + 6)^2 + (y + 15)^2 = 5. Therefore, the correct answer is Option 4.

Explanation

The equation of a circle with center (h, k) and radius r is (x - h)^2 + (y - k)^2 = r^2. In this case, the center is (-6, -15) and the radius is the square root of 5. Plugging these values into the equation, we get (x + 6)^2 + (y + 15)^2 = 5. Therefore, the correct answer is Option 4.

6. Which of the following is the equation of the graph below?

Option 1

Option 2

Option 3

Option 4

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Explanation

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7. What is the equation of the circle given that the center is (3,-2) and radius is 4?

Option 1

Option 2

Option 3

Option 4

The equation of a circle with center (h,k) and radius r is (x-h)^2 + (y-k)^2 = r^2. In this case, the center is (3,-2) and the radius is 4. So the equation of the circle is (x-3)^2 + (y+2)^2 = 4^2.

Explanation

The equation of a circle with center (h,k) and radius r is (x-h)^2 + (y-k)^2 = r^2. In this case, the center is (3,-2) and the radius is 4. So the equation of the circle is (x-3)^2 + (y+2)^2 = 4^2.

8. Which of the following is the center and the radius of the graph below?

Center (0,0), radius=3

Center (0,0) radius =6

Center (0,0), radius= 12

Center (0,0), radius = 36

The correct answer is center (0,0) radius =6. This means that the center of the graph is at the point (0,0) and the radius of the graph is 6 units. This implies that the graph is a circle with its center at the origin (0,0) and a radius of 6 units.

Explanation

The correct answer is center (0,0) radius =6. This means that the center of the graph is at the point (0,0) and the radius of the graph is 6 units. This implies that the graph is a circle with its center at the origin (0,0) and a radius of 6 units.

9. What is the equation of the circle given this graph?

Option 1

Option 2

Option 3

Option 4

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Explanation

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10. This is a curve formed by the intersection of a plane and a double right circular cone.

Circle

Conic

Parabola

Generator of the cone

The given statement describes a curve formed by the intersection of a plane and a double right circular cone. This curve is known as a conic. A conic can take various forms depending on the angle at which the plane intersects the cone. It can be a circle, an ellipse, a parabola, or a hyperbola. Therefore, the correct answer is conic.

Explanation

The given statement describes a curve formed by the intersection of a plane and a double right circular cone. This curve is known as a conic. A conic can take various forms depending on the angle at which the plane intersects the cone. It can be a circle, an ellipse, a parabola, or a hyperbola. Therefore, the correct answer is conic.

Conic Sections And Circle