Classify the conic section . Write its equation in standard form. 4x^2-y+3=0?
Solve for Y:
STD Form: Y = 4x^2 + 3.
This is a Y-parabola that opens upward.
V(0,3).
To classify the conic section and write its equation in standard form, we need to analyze the equation 4x^2 - y + 3 = 0.
The standard form equation for conic sections is generally written as Ax^2 + Bxy + Cy^2 + Dx + Ey + F = 0, where A, B, C, D, E, and F are constants.
Comparing this with our given equation, 4x^2 - y + 3 = 0, we can see that the coefficient of x^2 is 4 (A = 4), the coefficient of xy is 0 (B = 0), the coefficient of y^2 is -1 (C = -1), the coefficient of x is 0 (D = 0), the coefficient of y is -1 (E = -1), and the constant is 3 (F = 3).
Since B = 0, we can conclude that the conic section is not a parabola. Moreover, since A and C have opposite signs (A > 0 and C < 0), we can determine that the conic section is an ellipse.
Now, let's rewrite the equation in standard form. Starting with 4x^2 - y + 3 = 0, we move the constant term to the other side of the equation:
4x^2 - y = -3
We can rearrange the equation to highlight the coefficients:
4x^2 - y + 0x + 0y = -3
Grouping the variables:
4x^2 - y + 0x + 0y = -3
Now, we have the equation in standard form for an ellipse, with A = 4, B = 0, C = -1, D = 0, E = 0, and F = -3.