Assertion :Locus of z satisfying the equation $| z - 1 | + | z - 8 | = 5$ is an ellipse Reason: Sum of focal distances of any point on ellipse is constant.

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 and Reason is correct

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Solution

The correct option is D Assertion is incorrect and Reason is correct
Let $z = x + i y$
Therefore
$| z - 1 | + | z - 8 | = 5$
$\sqrt{y^{2} + (x - 1)^{2}} + \sqrt{y^{2} + (x - 8)^{2}} = 5$
$\sqrt{y^{2} + (x - 1)^{2}} = 5 - \sqrt{y^{2} + (x - 8)^{2}}$
$y^{2} + (x - 1)^{2} = 25 + y^{2} + (x - 8)^{2} - 10 \sqrt{y^{2} + (x - 8)^{2}}$
$10 \sqrt{y^{2} + (x - 8)^{2}} = 25 + (x - 8)^{2} - (x - 1)^{2}$
$10 \sqrt{y^{2} + (x - 8)^{2}} = 25 - 7 (2 x - 9)$
$10 \sqrt{(y^{2} + (x - 8)^{2}} = 25 - 14 x + 63$
$10 \sqrt{y^{2} + (x - 8)^{2}} = 88 - 14 x$
$5 \sqrt{y^{2} + (x - 8)^{2}} = 44 - 7 x$
$25 (y^{2} + (x - 8)^{2}) = 176 + 49 x^{2} - 616 x$
$25 y^{2} + 25 x^{2} - 400 x + 1600 = 176 - 616 x + 49 x^{2}$
$24 x^{2} - 25 y^{2} - 216 x = 1424$
This represents the equation of an hyperbola.

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