Correct Answer - Option 4 : parabola
Concept:
The general equation of conics of a second degree is given by:\(a{x^2} + 2hxy + b{y^2} + 2gx + 2fy + c = 0\)
and discriminant
Δ = abc + 2fgh – af2 – bg2 – ch2.
The above given equation represents a non-degenerate conics whose nature is given below in the table:
|
S.No
|
Condition
|
Nature of Conic
|
|
1
|
Δ ≠ 0, h = 0, a = b, e = 0
|
Circle
|
|
2
|
Δ ≠ 0, h2 – ab = 0, e = 1
|
Parabola
|
|
3
|
Δ ≠ 0, h2 – ab < 0, e < 1
|
Ellipse
|
|
4
|
Δ ≠ 0, h2 – ab > 0, e > 1
|
Hyperbola
|
|
5
|
Δ ≠ 0, h2 – ab > 0, a + b = 0, e = 1/2
|
Rectangular Hyperbola
|
Calculation:
The equation
16x2 - 24xy + 9y2 - 104x - 172y + 44 = 0
a = 16, b = 9, h = -12, g = = - 52, f = - 86, c = 44
∵ Δ = abc - 2fgh - af2 - bg2 - ch2
⇒ Δ = 16.9.44 - 2.(-86)(-52)(-12) - 16.(-86)2 - 9.(-52)2 - 44(-12)2
⇒ Δ = 6336 + 107328 - 118336 - 24336 - 6336
⇒ Δ = - 35344 ≠ 0
h2 - ab = (-12)2 - 16.9
⇒ h2 - ab = 0
As dicsueed above, we know that,
if Δ ≠ 0 and h2 - ab = 0 it represents parabola.
Hence, given equation reprtesents parabola.
