Classify the conic section
4x^2+5y^2-16x-30y+41=0 as a circle, ellipse, hyperbola, or parabola?
Ellipse
4(x^2 -4x +4) + 5(y^2 -6y +9) = -41 + 16 + 45
= 20
= 4(x-2)^2 + 5(y-3)^2
(x-2)^2/5 + (y-3)^2/4 = 1
The sum of 16terms of an A.P IS -504,while the sum of its 9terms is -126. Find the sum of its 30terms
To determine the classification of the conic section given by the equation: 4x^2 + 5y^2 - 16x - 30y + 41 = 0, we can start by examining the equation in standard form.
The standard form of an equation for a conic section is given by:
Ax^2 + Bxy + Cy^2 + Dx + Ey + F = 0
Comparing this to the given equation, we can identify the coefficients:
A = 4
B = 0 (since there is no xy term)
C = 5
D = -16
E = -30
F = 41
Next, we can calculate a quantity called the discriminant, which is given by the formula:
Discriminant (D) = B^2 - 4AC
In the case of a conic section, the discriminant can help us determine the type of conic section:
1. If D > 0 and A = C, then it is an ellipse.
2. If D = 0 and A = C, then it is a parabola.
3. If D < 0 and A ≠ C, then it is a hyperbola.
4. If D = 0 and A ≠ C, then it is a degenerate (special case) conic section.
Let's calculate the discriminant for the given equation:
D = B^2 - 4AC
D = 0^2 - 4 * 4 * 5
D = 0 - 80
D = -80
Since the discriminant (D) is negative (D < 0) and the coefficients of x^2 (4) and y^2 (5) are not equal, the conic section given by the equation: 4x^2 + 5y^2 - 16x - 30y + 41 = 0 is a hyperbola.