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Distinguishing between Conics from General Equation and Eccentricity

The values of...

Question

The value(s) of $k$ if the equation $9 x^{2} + 4 y^{2} + 2 k x y + 4 x - 2 y + 3 = 0$ represents a parabola, is (are)

6

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36

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- 6

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- 36

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Solution

The correct option is C $- 6$
The general equation of second degree is :
$a x^{2} + 2 h x y + b y^{2} + 2 g x + 2 f y + c = 0$
Given: $9 x^{2} + 4 y^{2} + 2 k x y + 4 x - 2 y + 3 = 0$
Comparing both,
$a = 9, b = 4, c = 3, h = k, g = 2, f = - 1$
For equation to be a parabola,
$h^{2} = a b$ and $Δ \neq 0$
$\Rightarrow k^{2} - 9 \times 4 = 0$
$\Rightarrow k = \pm 6$

Now, $Δ = a b c + 2 f g h - a f^{2} - b g^{2} - c h^{2}$
$= 108 - 4 k - 9 - 16 - 3 k^{2}$
$= - 3 k^{2} - 4 k + 83 \neq 0$
For $k = 6$ or $k = - 6$

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