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  • The set of value of , for which the circle  <img alt="\mathrm{C: 4 x^{2}+4 y^{2}-12 x+8 y+k=0}" s

The set of value of \mathrm{k}, for which the circle  \mathrm{C: 4 x^{2}+4 y^{2}-12 x+8 y+k=0} lies inside the fourth quadrant and the point \mathrm{\left ( 1,-\frac{1}{3} \right )} lies on or inside the circle \mathrm{C}, is

Option: 1

an empty set


Option: 2

\mathrm{\left ( 6,\frac{65}{9} \right ]}


Option: 3

\mathrm{\left [ \frac{80}{9},10 \right )}


Option: 4

\mathrm{\left ( 9,\frac{92}{9} \right ]}


Answers (1)

best_answer

\mathrm{Cirde: x^{2}+y^{2}-3 x+2 y+\frac{k}{4}=0}\\

\mathrm{Centre: \left(\frac{3}{2},-1\right) radius: \sqrt{\left(\frac{3}{2}\right)^{2}+(1)^{2}-\frac{k}{4}}}

Radius must be lies than 1 for the circle to lie in the fourth quadrant.

\mathrm{\Rightarrow \quad \frac{9}{4}+1-\frac{k}{4}<1 \Rightarrow k>9}          .........(1)

Point \mathrm{\left ( 1,-\frac{1}{3} \right )} lies on or inside the circle

\mathrm{\Rightarrow s_{1} \leqslant 0 \Rightarrow 1+\frac{1}{9}-3-\frac{2}{3}+\frac{k}{4} \leqslant 0 \Rightarrow \frac{k}{4} \leqslant \frac{8}{3}-\frac{1}{9}=\frac{23}{9}} \\

\mathrm{\Rightarrow k \leqslant \frac{92}{9}} \\               ..............(2)

From (1 ) and ( 2 )

\mathrm{k \in\left(9, \frac{92}{9}\right] }

Hence the correct answer is option 4.

 

Posted by

HARSH KANKARIA

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