Q.300. Test of a 2nd degree equation in R^2 to represent a particular conic.

Question 300.
If it is given that a second degree equation, ax^2 + 2hxy + by^2 + 2gx + 2fy + c = 0 represents a conic, determine conditions for it to represent a parabola, an ellipse or a hyperbola.

Answer 300.
The general second degree equation is
ax^2 + 2hxy + by^2 + 2gx + 2fy + c = 0 ... (1)

Let P (x, y) be any point on a conic having its focus S (p, q) and directrix lx + my + n = 0.
=> SP^2 = e^2 PM^2
=> [(x -p)^2 + (y - q)^2] = e^2 * (lx + my + n)^2 / (l^2 + m^2)
=> (l^2 + m^2) [(x -p)^2 + (y - q)^2] = e^2 * (lx + my + n)^2 ... (2)

Comparing eqns. (1) and (2),
a = l^2 + m^2 - e^2l^2
b = l^2 + m^2 - e^2m^2 and
2h = - 2lme^2

=> h^2 - ab
= l^2m^2e^4 - (l^2 + m^2 - e^2l^2) (l^2 + m^2 - e^2m^2)
= e^2 (l^2 + m^2)^2 - (l^2 + m^2)^2
= (l^2 + m^2)^2 (e^2 - 1)

Thus,
h^2 - ab < 0 <=> e < 1 <=> conic is an ellipse,
h^2 - ab = 0 <=> e = 1 => conic is a parabola and
h^2 - ab > 0 <=> e > 1 => conic is a hyperbola,

Source(s):
Modern Mathematics written by Rev. Father C. G. Valles, Prof. S. C. Vora and Dr. J. D. Acharya and published by C. Jamnadas & Co. Ahmedabad, Gujarat, India.

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