Conic Sections And Circles

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 correct answer is Option 2.

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 (3, -2) and the radius is 4. Plugging these values into the equation, we get (x-3)^2 + (y+2)^2 = 16, which is the equation of the circle.

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The equation of a circle with radius r and center at the origin (0,0) is given by x^2 + y^2 = r^2. In this case, the radius is 8, so the equation of the circle is x^2 + y^2 = 64.

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A circle is a set of all coplanar points that are equidistant from a fixed point, known as the center. The distance from any point on the circle to the center remains constant, which defines the shape of a circle. This property distinguishes a circle from other conic sections such as the parabola, hyperbola, and ellipse, which have different characteristics and equations.

The standard form of a quadratic equation is given by Ax^2 + By^2 + Cx + Dy + E = 0, where A, B, C, D, and E are constants. In the given equation x^2 + y^2 + 14x - 12y + 4 = 0, we can rearrange the terms to match the standard form. By grouping the x-terms and the y-terms separately, we get (x^2 + 14x) + (y^2 - 12y) + 4 = 0. Completing the square for both x and y, we have (x^2 + 14x + 49) + (y^2 - 12y + 36) + 4 - 49 - 36 = 0. Simplifying further, we get (x + 7)^2 + (y - 6)^2 - 81 = 0. Therefore, the standard form of the equation is (x + 7)^2 + (y - 6)^2 = 81, which matches with Option 1.

The correct answer is center (3, 13) radius=4. This is because the center coordinates match the given center (-3, 13) and the radius value is also 4, which matches the given radius value.

The correct answer is Center (0, 0) , radius = 6. This is because the graph is centered at the point (0, 0) and the distance from the center to any point on the graph is 6 units.

The curve formed by the intersection of a plane and a double circular right cone is known as a conic. A conic is a general term that includes various types of curves such as circles, ellipses, parabolas, and hyperbolas. In this case, since the cone is double circular, the resulting curve would be a conic. Therefore, the correct answer is conic.