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  • Let the circle  <img alt="\mathrm{S: 36 x^2+36 y^2-108 x+120 y+C=0}" src="https://entrancecorner.oncodecogs.com/gif.latex?%5Cmathrm%7BS%3A%2036%20x%5E2&plus;36%20y%5E2-108%20x&plu

Let the circle  \mathrm{S: 36 x^2+36 y^2-108 x+120 y+C=0}  be such that it neither intersects nor touches the co-ordinate axes. If the point of intersection of the lines, \mathrm{x-2 y=4} and \mathrm{2 x-y=5} lies inside the circle S, then

Option: 1

\mathrm{81<C<156}


Option: 2

\mathrm{100<C<165}


Option: 3

\mathrm{100<C<156}


Option: 4

\mathrm{\frac{25}{9}<C<\frac{13}{3}}


Answers (1)

best_answer

Given, circle S is
\mathrm{ 36 x^2+36 y^2-108 x+120 y+C=0 }
\mathrm{\Rightarrow x^2+y^2=\frac{108 x}{36}-\frac{120 y}{36}-\frac{C}{36} }
\mathrm{\Rightarrow x^2+y^2=3 x-\frac{10}{3} y-\frac{C}{36} }
\mathrm{\text { or } x^2+y^2-3 x+\frac{10}{3} y+\frac{C}{36}=0 }

Given that circle does not touch the co-ordinate axes.
\mathrm{ \Rightarrow g^2-c<0 \text { and } f^2-c<0 \\ }
\mathrm{ \Rightarrow\left(-\frac{3}{2}\right)^2-\frac{C}{36}<0 \text { and }\left(\frac{10}{6}\right)^2-\frac{C}{36}<0 \\ }
\mathrm{ {\left[\because \text { Here } g=\frac{-3}{2}, f=\frac{10}{6} \text { and } c=\frac{C}{36}\right]} \\ }
\mathrm{ \Rightarrow \frac{9}{4}-\frac{C}{36}<0 \text { and } \frac{100}{36}-\frac{C}{36}<0 \\ }
\mathrm{ \Rightarrow \frac{9}{4}<\frac{C}{36} \text { and } \frac{100}{36}<\frac{C}{36} \\ }
\mathrm{ \Rightarrow C>\frac{9 \times 36}{4} \text { and } C>\frac{36 \times 100}{36} \\ }
\mathrm{ \Rightarrow C>81 }   .....(i)   and  \mathrm{C>100 }    .....(ii)

Now, as \mathrm{g^2+f^2-c>0 }
\mathrm{ \therefore\left(-\frac{3}{2}\right)^2+\left(\frac{5}{3}\right)^2-\frac{C}{36}>0 \\ }
\mathrm{ \Rightarrow \frac{C}{36}<\frac{9}{4}+\frac{25}{9} \Rightarrow C<181 }    .....(iii)

Intersection point of 2x - y = 5 and x - 2y = 4 is (2,-1) and (2,-1) lies inside the circle. So, S(2,-1) < 0
\mathrm{ \therefore \quad 36(2)^2+36(-1)^2-108(2)+120(-1)+C<0 \\ }
\mathrm{ \Rightarrow C<156 \\ }             .....(iv)
\mathrm{ \therefore } By (ii), (iii) and (iv), we have 
\mathrm{ 100<C<156 . }

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chirag

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