the equation 4x^2-12xy+9y^2=0 represents
Answers
Therefore the given equation 4x² - 12xy + 9y² = 0 represents a 'Parabola'.
Given:
The equation: 4x² - 12xy + 9y² = 0
To Find:
The curve represented by the given equation 4x² - 12xy + 9y² = 0.
Solution:
The given question can be solved very easily as shown below.
Given equation: 4x² - 12xy + 9y² = 0
Concept:
Let us assume the given second degree equation be Ax² + Bxy + Cy² + Dx + Ey + F = 0, then the discriminant is given by B² - 4AC.
The conditions for different conics are given as follows,
⇒ For circle, B² - 4AC < 0 and B = 0 and A = C
⇒ For ellipse, B² - 4AC < 0 and B ≠ 0 and A ≠ C
⇒ For parabola, B² - 4AC = 0
⇒ For hyperbola, B² - 4AC > 0
Now comparing the given equation with the standard form.
⇒ A = 4, B = -12, and C = 9
Then B² - 4AC = (-12)² - 4(4)(9) = 144 - 144 = 0
So the discriminant B² - 4AC = 0, Hence the given equation is Parabola.
Therefore the given equation 4x² - 12xy + 9y² = 0 represents a 'Parabola'.
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EXPLANATION.
The equation 4x² - 12xy + 9y² = 0 represents.
As we know that,
Concepts :
A pair of straight lines through origin.
A homogeneous equation of degree two of the type ax² + 2hxy + by² = 0 always represents a pair of straight lines passing through the origin and if,
(a) h² > ab ⇒ lines are real and distinct.
(b) h² = ab ⇒ lines are coincident.
(c) h² < ab ⇒ lines are imaginary with real point of intersections that is (0,0).
Using this concepts in this question, we get.
⇒ 4x² - 12xy + 9y² = 0. - - - - - (1).
⇒ ax² + 2hxy + by² = 0. - - - - - (2).
Comparing equation (1) and equation (2), we get.
⇒ a = 4.
⇒ 2h = - 12.
⇒ h = - 6.
⇒ b = 9.
We get : a = 4, b = 9, h = - 6.
⇒ h² = (-6)² = 36.
⇒ h² = 36.
⇒ ab = 4 x 9 = 36.
⇒ ab = 36.
∴ h² = ab ⇒ lines are coincident.
MORE INFORMATION.
General equation of second degree representing a pair of straight lines :
ax² + 2hxy + by² + 2gx + 2fy + c = 0 represents a pair of straight lines if,
abc + 2fgh - af² - bg² - ch² = 0.
The angle θ between the two lines representing by a general equation is the same as that between the two lines representing by its homogeneous part only.