The equation x210−k+y2k−4=1represents an ellipse if

x210−k+y2k−4=1⇒10−k>0⇒k 0,⇒k>4,10−k≠k−4∴k≠7⇒k∈(4,7)∪(7,10)

The equation x210−k+y2k−4=1represents an ellipse if

x210−k+y2k−4=1⇒10−k>0⇒k 0,⇒k>4,10−k≠k−4∴k≠7⇒k∈(4,7)∪(7,10)

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The equation $\frac{x^{2}}{10 - k} + \frac{y^{2}}{k - 4} = 1$ represents an ellipse if

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k < 4

k > 10

4 < k < 10

$k \in (4, 7) \cup (7, 10)$

answer is D.

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Detailed Solution

$\begin{array}{l} \frac{x^{2}}{10 - k} + \frac{y^{2}}{k - 4} = 1 \Rightarrow 10 - k > 0 \Rightarrow k < 10, \\ k - 4 > 0, \Rightarrow k > 4, \\ 10 - k \neq k - 4 ∴ k \neq 7 \\ \Rightarrow k \in (4, 7) \cup (7, 10) \end{array}$

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The equation x210−k+y2k−4=1represents an ellipse if

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The equation $\frac{x^{2}}{10 - k} + \frac{y^{2}}{k - 4} = 1$ represents an ellipse if