A) \[20{{r}^{2}}\]
B) \[52{{r}^{2}}\]
C) \[\frac{52}{9}{{r}^{2}}\]
D) \[\frac{20}{9}{{r}^{2}}\]
Correct Answer: B
Solution :
Equation of line is \[3x-2y=k\] ...... (i) Circle is \[{{x}^{2}}+{{y}^{2}}=4{{r}^{2}}\] ..... (ii) Equation of line can be written as \[y=\frac{3}{2}x-\frac{k}{2}\] Here, \[c=-\frac{k}{2},\,m=\frac{3}{2}\] Now the line will meet the circle at one point, if \[c=\pm a\sqrt{1+{{m}^{2}}}\] \[=\frac{-k}{2}=\pm (2r)\,\sqrt{1+{{\left( \frac{3}{2} \right)}^{2}}}\] {From (ii), a = 2r} \[=\frac{{{k}^{2}}}{4}=4{{r}^{2}}\times \frac{13}{4}\], \[\therefore \] \[{{k}^{2}}=52{{r}^{2}}.\]
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