The number of integral values of $k$ for which the line, $3 x + 4 y = k$ intersects the circle, $x^{2} + y^{2} - 2 x - 4 y + 4 = 0$ at two distinct points is

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Solution

$C : (1, 2) & r = 1$
$| C P | < r$
$\Rightarrow ∣ ∣ ∣ \frac{3 \cdot 1 + 4 \cdot 2 - k}{5} ∣ ∣ ∣ < 1$
$\Rightarrow | 11 - k | < 5$
$\Rightarrow - 5 < k - 11 < 5$
$\Rightarrow 6 < k < 16$
$\Rightarrow k = 7, 8, 9, \dots, 15$
$\Rightarrow$ total $9$ value of $k$

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