Conic Sections And Introduction To Circles

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 equation represents all the points (x,y) that are 5 units away from the center (0,0), forming a circle with a radius of 5.

Explanation

The equation of a circle with center (0,0) and radius 5 is x^2 + y^2 = 25. This equation represents all the points (x,y) that are 5 units away from the center (0,0), forming a circle with a radius of 5.

2) This is a set of coplanar points such as the distance from a fixed point is constant.

Circle

Parabola

Hyperbola

Ellipse

A circle is a set of coplanar points where the distance from a fixed point (called the center) is constant. The distance between any point on the circle and the center is always the same, which makes it a perfect fit for the given description.

Explanation

A circle is a set of coplanar points where the distance from a fixed point (called the center) is constant. The distance between any point on the circle and the center is always the same, which makes it a perfect fit for the given description.

3) 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. Therefore, the equation of the circle is (x-3)^2 + (y+2)^2 = 16.

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. Therefore, the equation of the circle is (x-3)^2 + (y+2)^2 = 16.

4) What is the equation of the circle whose center is (-6,-15) and a radius is square root of 5.

Option 1

Option 2

Option 3

Option 4

The equation of a circle with center (h,k) and radius r is given by (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 equation of the circle is (x+6)^2 + (y+15)^2 = 5.

Explanation

The equation of a circle with center (h,k) and radius r is given by (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 equation of the circle is (x+6)^2 + (y+15)^2 = 5.

5) 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 x2 + y2 + 14x - 12y + 4 = 0 is (x + 7)2 + (y - 6)2 = 25. This is because the standard form of the equation of a circle is (x - h)2 + (y - k)2 = r2, where (h, k) is the center of the circle and r is the radius. In this case, the center of the circle is (-7, 6) and the radius is 5.

Explanation

The standard form of the equation x2 + y2 + 14x - 12y + 4 = 0 is (x + 7)2 + (y - 6)2 = 25. This is because the standard form of the equation of a circle is (x - h)2 + (y - k)2 = r2, where (h, k) is the center of the circle and r is the radius. In this case, the center of the circle is (-7, 6) and the radius is 5.

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 given circle based on the graph?

Option 1

Option 2

Option 3

Option 4

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Explanation

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8) 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 coordinates (-3, 13) and (3, 13) are the only ones that have the same y-coordinate, which matches the given equation. Additionally, the radius of 4 matches the given equation.

Explanation

The correct answer is center (3, 13) radius= 4. This is because the center coordinates (-3, 13) and (3, 13) are the only ones that have the same y-coordinate, which matches the given equation. Additionally, the radius of 4 matches the given equation.

9) 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 is a type of curve that includes circles, ellipses, parabolas, and hyperbolas. Since the question asks for the correct answer, the term "conic" accurately represents the curve formed by the intersection of the plane and the double right circular cone.

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 is a type of curve that includes circles, ellipses, parabolas, and hyperbolas. Since the question asks for the correct answer, the term "conic" accurately represents the curve formed by the intersection of the plane and the double right circular cone.