A nondegenerate conic section in the form 𝐴𝑥2+𝐶𝑦2+𝐷𝑥+𝐸𝑦+𝐹=0, in which 𝐴 and 𝐶 are not both zero, is a/an _________ if 𝐴=𝐶, a/an ________ if 𝐴𝐶=0, a/an _________ if 𝐴≠𝐶 and 𝐴𝐶>0, and a/an ________ if 𝐴𝐶<0.Group of answer choiceshyperbola; circle; parabola; ellipsecircle; parabola; ellipse; hyperbolaellipse; hyperbola; circle; parabolaparabola; ellipse; hyperbola; circle
Question
A nondegenerate conic section in the form 𝐴𝑥2+𝐶𝑦2+𝐷𝑥+𝐸𝑦+𝐹=0, in which 𝐴 and 𝐶 are not both zero, is a/an _________ if 𝐴=𝐶, a/an ________ if 𝐴𝐶=0, a/an _________ if 𝐴≠𝐶 and 𝐴𝐶>0, and a/an ________ if 𝐴𝐶<0.Group of answer choiceshyperbola; circle; parabola; ellipsecircle; parabola; ellipse; hyperbolaellipse; hyperbola; circle; parabolaparabola; ellipse; hyperbola; circle
Solution
The correct answer is: circle; parabola; ellipse; hyperbola.
Here's why:
-
If A = C, the conic section is a circle. This is because the coefficients of the squared terms are equal, creating a symmetrical shape.
-
If AC = 0, the conic section is a parabola. This is because one of the squared terms is missing, creating a U-shaped curve.
-
If A ≠ C and AC > 0, the conic section is an ellipse. This is because the coefficients of the squared terms are different, creating an elongated circle.
-
If AC < 0, the conic section is a hyperbola. This is because the product of the coefficients of the squared terms is negative, creating two separate curves.
Similar Questions
How can you distinguish parabolas from other conic sections by looking at their equations without an 𝑥𝑦−term?Group of answer choicesIn equations for parabolas, 𝑥 and 𝑦 are both squared and the coefficients on them are the same, including the sign.In equations for parabolas, 𝑥 and 𝑦 are both squared, and exactly one of the coefficients is negative and exactly one of the coefficients is positive.In equations for parabolas, 𝑥 and 𝑦 are both squared, and the coefficients are positive but different.In equations for parabolas, either 𝑥 and 𝑦 is squared, but not both.
The point, line, and pair of intersecting lines are special types of conic sections.A.TrueB.False
A conic section whose eccentricity is less than one is known as:
Which is the best definition of a conic section?
When a right circular cone is cut by a planeparallel to the axis of the cone the conicsection formed is a/an _____________.A. HyperbolaB. ParabolaC. CircleD. Ellipse
Upgrade your grade with Knowee
Get personalized homework help. Review tough concepts in more detail, or go deeper into your topic by exploring other relevant questions.