the equation 2x^2+ cy^2+Dx+Ey+f= 0
represent a conic. state the value(s) of C for which each of the following are possible:
A circle:
My answer: c= 2
an ellipse:
My answer: c>0, cant equal 2
Parabola:
My answer:c=0
a hyperbola:
My answer: c<o ( any negative number)
are these right thanks!
You are exactly correct!
You are right on the money nice job
Yes, your answers are correct!
To determine the type of conic that the equation represents, we examine the coefficient of the y^2 term (C). Here, we analyze the possible values of C for each type of conic:
1. Circle:
A circle is formed when the x^2 and y^2 terms have the same coefficient. In this equation, the coefficient of the x^2 term is already 2, so for it to form a circle, the coefficient of the y^2 term (C) should also be 2.
Therefore, for a circle: C = 2.
2. Ellipse:
An ellipse is formed when the coefficients of the x^2 and y^2 terms have different positive values. In this equation, C should have a positive value, but it cannot be equal to 2 because that would make it a circle.
Therefore, for an ellipse: C > 0 (but not equal to 2).
3. Parabola:
A parabola is formed when the coefficient of the y^2 term is zero. In this equation, we can see that there is no y^2 term present. Hence, the coefficient of the y^2 term (C) should be zero.
Therefore, for a parabola: C = 0.
4. Hyperbola:
A hyperbola is formed when the coefficients of the x^2 and y^2 terms have different signs. In this equation, the coefficient of the x^2 term is positive (2), so for it to form a hyperbola, the coefficient of the y^2 term (C) should be negative.
Therefore, for a hyperbola: C < 0 (negative values).
You have correctly identified the values of C for each type of conic: circle (C = 2), ellipse (C > 0 but not 2), parabola (C = 0), and hyperbola (C < 0). Well done!