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

Option: 1

9


Option: 2

6


Option: 3

8


Option: 4

-9


Answers (1)

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If the line cuts the circle, then CP < r................(i)
Since, equation of circle is \mathrm{x^2+y^2-2 x-4 y+4=0} 

\mathrm{\therefore \quad} Centre =(1,2) and radius \mathrm{=\sqrt{1+4-4}=1} 

Also, the equation of line is  \mathrm{3 x+4 y=k}

\mathrm{ \therefore \text { Using (i), }\left|\frac{3+8-k}{\sqrt{9+16}}\right|<1 \\ }

\mathrm{ \Rightarrow\left|\frac{11-k}{5}\right|<1 \Rightarrow|11-k|<5 \\ }

\mathrm{ \Rightarrow 11-k<5 \text { and }-(11-k)<5 \\ }

\mathrm{ \Rightarrow k>6 \text { and }-11+k<5 \Rightarrow k<16 \\ }

\mathrm{ \Rightarrow k \in(6,16) }
Integral values of k=7,8,9,10,11,12,13,14,15 

Thus, number of integral values of k is 9 .

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Ritika Kankaria

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