The equation $3 x^{2} + 4 y^{2} - 18 x + 16 y + 43 = k$

represents empty set, if

k < 0

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represents an ellipse, if

k > 0

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represents a point, if

k = 0

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cannot represent a real pair of straight lines for any value of

k

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Solution

The correct option is D cannot represent a real pair of straight lines for any value of $k$
$3 x^{2} + 4 y^{2} - 18 x + 16 y + 43 = k \Rightarrow 3 (x - 3)^{2} + 4 (y + 2)^{2} = k$

If $k < 0$ then no locus.

If $k > 0$
$\frac{(x - 3)^{2}}{k / 3} + \frac{(y + 2)^{2}}{k / 4} = 1$
Locus is an ellipse.

If $k = 0$
$3 (x - 3)^{2} + 4 (y + 2)^{2} = 0 \Rightarrow x = 3, y = - 2$
Locus is a point.

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Similar questions

3 x^{2} + 4 y^{2} - 18 x + 16 y + 43 = k

3 x^{2} + 4 y^{2} - 18 x + 16 y + 43 = c

x^{2} + y^{2} - 2 x - 4 y - 20 = 0

The equation of the circle whose radius is 5 and which touches the circles $x^{2} + y^{2} - 2 x - 4 y - 20 = 0$ externally at the point (5,5), is

3 x^{2} = 16 y

4 y^{2} = 9 x

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