# The number of integral values of k for which the line, $3x+4y=k$ intersects the circle, $x^{2}+y^{2}-2x-4y+4=0$ at two distinct point is_______. Option: 1 4 Option: 2 9 Option: 3 14 Option: 4 21

$\\\text { Circle } x^{2}+y^{2}-2 x-4 y+4=0 \\ \Rightarrow(x-1)^{2}+(y-2)^{2}=1$

Center = (1,2) ; Radius = 1

Line 3x + 4y – k = 0 intersects the circle at two distinct points.

distance of centre from the line < radius

\begin{aligned} &\Rightarrow\left|\frac{3 \times 1+4 \times 2-\mathrm{k}}{\sqrt{3^{2}+4^{2}}}\right|<1\\ &\Rightarrow|11-k|<5\\ &\Rightarrow 6<\mathrm{k}<16\\ &\Rightarrow \mathrm{k} \in\{7,8,9, \ldots \ldots 15\} \text { since } \mathrm{k} \in \mathrm{I}\\ &\text { Number of } \mathrm{K} \text { is } 9 \end{aligned}

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