The set of values of 'c' so that the equations $y = | x | + c a n d x^{2} + y^{2} - 8 | x | - 9 = 0$ have no solution is

$(\infty, - 3) \cup (3, \infty)$

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$(- 3, 3)$

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$(- \infty, 5 \sqrt{2}) \cup (5 \sqrt{(} 2), \infty)$

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$5 \sqrt{2} - 4, \infty$

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Solution

The correct option is D
$5 \sqrt{2} - 4, \infty$

Since $y = | x | + c a n d x^{2} + y^{2} - 8 | x | - 9 = 0$ both are symmetrically about y-axis for x>0, y=x+c.
Equation of tangent to circle $x^{2} + y^{2} - 8 x - 9 = 0$ parallel to y=x+c is $y = (x - 4) + 5 \sqrt{1 + 1}$
$\Rightarrow y = x + (5 \sqrt{2} - 4)$
For no solution c> $5 \sqrt{2} - 4$
$∴ c ϵ (5 \sqrt{2} - 4, \infty)$

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