To determine what type of conic section a general second-degree equation represents, we must compare it with the standard general equation and analyze its discriminant elements.
Step 1: Compare with the General Second-Degree Equation
The general second-degree equation in two variables is given by:
ax2+2hxy+by2+2gx+2fy+c=0
Now, compare this standard form with the given equation:
3x2+10xy+11y2+14x+12y+5=0
By comparing the corresponding coefficients, we extract the following values:
a=3
2h=10⟹h=5
b=11
2g=14⟹g=7
2f=12⟹f=6
c=5
Step 2: Check for a Non-Degenerate Conic (Δ=0)
Before classifying the curve, we must ensure that the equation does not represent a degenerate conic (like a pair of straight lines). We do this by calculating the determinant (Δ):
Δ=abc+2fgh−af2−bg2−ch2
Substitute our extracted values into the formula:
Δ=(3)(11)(5)+2(6)(7)(5)−3(6)2−11(7)2−5(5)2
Δ=165+420−3(36)−11(49)−5(25)
Δ=585−108−539−125
Δ=585−772=−187
Since Δ=−187=0, the equation represents a non-degenerate conic section.
Step 3: Evaluate the Discriminant Value (h2−ab)
To classify the type of non-degenerate conic section, we check the sign of the value h2−ab:
Condition | Nature of Conic |
h^2 - ab < 0 | Ellipse (or Circle if a=b and h=0) |
h2−ab=0 | Parabola |
h^2 - ab > 0 | Hyperbola |
Let's compute h2−ab with our values:
h2−ab=(5)2−(3)(11)
h2−ab=25−33
h2−ab=−8
Since -8 < 0, we have:
h^2 - ab < 0
This strictly satisfies the mathematical condition for an ellipse.
