Assertion :Statement - 1: If the foci of an hyperbola are at points $(4, 1)$ and $(- 6, 1)$ , eccentricity is $\frac{5}{4}$ , then the length of the transverse axis is $4$ . Reason: Statement - 2: Distance between the foci of a hyperbola is equal to the product of its eccentricity and the length of the transverse axis.

Both Assertion and Reason are correct and Reason is the correct explanation for Assertion

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Both Assertion and Reason are correct but Reason is not the correct explanation for Assertion

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Assertion is correct but Reason is incorrect

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Assertion is incorrect but reason is correct

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Solution

The correct option is D Assertion is incorrect but reason is correct
Here distance between focus is $S S^{'} = \sqrt{(4 + 6)^{2} + (1 - 1)^{2}} = 10, e = \frac{5}{4}$
We know in hyperbola distance between foci $= 2 a e = 2 \times a \times \frac{5}{4} = 10 \Rightarrow a = 4$ . Thus length of transverse axis is $2 a = 8$
Hence Assertion is incorrect.

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Assertion :Statement - 1: If the foci of an hyperbola are at points (4,1) and (−6,1), eccentricity is 54, then the length of the transverse axis is 4. Reason: Statement - 2: Distance between the foci of a hyperbola is equal to the product of its eccentricity and the length of the transverse axis.

The correct option is D Assertion is incorrect but reason is correctHere distance between focus is SS′=√(4+6)2+(1−1)2=10,e=54We know in hyperbola distance between foci =2ae=2×a×54=10⇒a=4. Thus length of transverse axis is 2a=8Hence Assertion is incorrect.