Find the greatest value of $x$ satisfying the relation $x^{2} + y^{2} = 6 x - 8 y$ .

8

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9

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10

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12

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Solution

The correct option is A $8$
The given curve is, $x^{2} + y^{2} = 6 x - 8 y$
$\Rightarrow (x^{2} - 6 x) + (y^{2} + 8 y) = 0$
$\Rightarrow (x^{2} - 6 x + 9) + (y^{2} + 8 y + 16) = 9 + 16$ , make perfect square
$\Rightarrow (x - 3)^{2} + (y + 4)^{2} = 25 = 5^{2}$
Clearly above equation represents circle having centre $(3, 4)$ and radius $5$
So any parametric point the above circle is taken as
$x - 3 = 5 cos θ$ and $y + 4 = 5 cos θ$
$\Rightarrow x = 3 + 5 cos θ$
Now we know that $- 1 \leq cos θ \leq 1$
Hence $x_{m a x} = 3 + 5 \cdot 1 = 8$

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