General equation of a conic :
Ax2 + Bxy +Cy2 + Dx + Ey + F = 0
The graph of the second degree equation will be a circle or a parabola or an ellipse or a hyperbola.
(1) When A = C = 1, B = 0, D = -2h, E = −2k, F = h2 + k2 - r2, the general equation reduces to
(x − h)2 + ( y − k)2 = r2,
which is a circle.
(2) When B = 0 and either A or C = 0 , the general equation yields a parabola under study, at this level.
(3) When A ≠ C and A and C are of the same sign, the general equation yields an ellipse.
(4) When A≠C and A and C are of opposite signs, the general equation yields a hyperbola
(5) When A = C and B = D = E = F = 0 , the general equation yields a point x2 + y2 = 0 .
(6) When A = C = F and B = D = E = 0 , the general equation yields an empty set x2 + y2 +1 = 0 , as there is no real solution.
(7) When A ≠ 0 or C ≠ 0 and others are zeros, the general equation yield coordinate axes.
(8) When A = -C and rests are zero, the general equation yields a pair of lines x2 - y2 = 0
Example 1 :
Identify the type of conic section for each of the equations.
(i) 2x2 − y2 = 7
Solution :
By comparing the given equation with the general form of conic Ax2 + Bxy + Cy2 + Dx + Ey + F = 0, we get
A = 2, C = -1 and F = -7
The values of A and C are not equal and it has opposite signs. Hence it is a hyperbola.
(ii) 3x2 + 3y2 − 4x + 3y - 10 = 0
By comparing the given equation with the general form of conic Ax2 + Bxy + Cy2 + Dx + Ey + F = 0, we get
3x2 + 3y2 − 4x + 3y - 10 = 0
Dividing the entire equation by 3, we get
x2 + y2 − (4/3)x + y - (10/3) = 0
x2 − (4/3)x + y2 + y = 10/3
(x - (4/6))2 - (4/6)2 + (y + (1/2))2 - (1/2)2 = 10/3
(x - (2/3))2 + (y + (1/2))2 = 10/3 + (4/9) + (1/4)
(x - (2/3))2 + (y + (1/2))2 = (120 + 16 + 9)/36
(x - (2/3))2 + (y + (1/2))2 = 145/36
(x - (2/3))2 + (y + (1/2))2 = (√145/6)2
Hence it is a circle.
(iii) 3x2 + 2y2 = 14
Solution :
3x2 + 2y2 = 14
A ≠ C and A and C are of the same sign, the general equation yields an ellipse.
(iv) x2 + y2 + x − y = 0
Solution :
x2 + x + y2− y = 0
(x + (1/2))2 - (1/2)2 + (y - (1/2))2 - (1/2)2 = 0
(x + (1/2))2 + (y - (1/2))2 = 2/4
(x + (1/2))2 + (y - (1/2))2 = 1/2
Hence the given is the equation of circle.
(v) 11x2 − 25y2 − 44x + 50y − 256 = 0
Solution :
11x2 − 44x − 25y2 + 50y − 256 = 0
A = 11, B = 0, C = -25
A and C are not equal and they have opposite signs. It is equation of hyperbola.
(vi) y2 + 4x + 3y + 4 = 0
Solution :
B = 0 and A = 0. Hence it is the equation of parabola.
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