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What is the position of circle \mathrm{C_2: x^2+(y-1)^2=16} with respect to circle \mathrm{C_1: x^2+y^2-x=26} ?

Option: 1

\mathrm{C_1} lies completely inside \mathrm{C_2}


Option: 2

\mathrm{C_1} and \mathrm{C_2} intersect each other.


Option: 3

\mathrm{C_2} lies completely inside \mathrm{C_1}.


Option: 4

Both \mathrm{C_1} and \mathrm{C_2} touch externally


Answers (1)

If the two circles \mathrm{C_1} and \mathrm{C_2} touch internally then

\mathrm{C_1 C_2=r_1-r_2}. But if  \mathrm{C_2} lies inside the circle \mathrm{C_1} then \mathrm{C_1 C_2<r_1-r_2}.


Here \mathrm{C_1\left(\frac{1}{2}, 0\right), r_1=\frac{1}{2} \sqrt{105}, C_2=(0,1), r_2=4}
         \mathrm{ C_1 C_2=\sqrt{\frac{1}{4}+1}=\frac{1}{2} \sqrt{5} \text { and } r_1-r_2=\frac{1}{2} \sqrt{105}-4 }
Clearly \mathrm{ C_1 C_2<r_1-r_2 }. Hence circle \mathrm{ C_2 } lies completely inside circle \mathrm{ C_1 }.

Posted by

Sumit Saini

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