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  • Two circles <img alt="\mathrm{x^2+y^2-10 x+16=0 \text { and } x^2+y^2=r^2{}" src="https://entrancecorner.oncodecogs.com/gif.latex?%5Cmathrm%7Bx%5E2&plus;y%5E2-10%20x&plus;16%3D0%20%5Ct

Two circles \mathrm{x^2+y^2-10 x+16=0 \text { and } x^2+y^2=r^2{}  intersect each other at two distinct points if

Option: 1

\mathrm{r<2}


Option: 2

\mathrm{r>8}


Option: 3

\mathrm{2<r<8}


Option: 4

none of these


Answers (1)

best_answer

For circle \mathrm{x^2+y^2-10 x+16} , the centre is  \mathrm{C_2(5,0)} and \mathrm{r_2=\text { radius }=\sqrt{25-16}=3}

For circle \mathrm{x^2+y^2=r^2} , , the centre is  \mathrm{C_1(0,0)} and radius = r

The circles intersect at two distinct points if the distance between the centre is greater than the difference between their radii and less than the sum of their radii.

\mathrm{\begin{aligned} & \Rightarrow|3-r|<C_1 C_2<3+r \\ & \text { Or }|3-r|<5<3+r \end{aligned}}

\mathrm{r-3<5 \text { or } r<8}

Posted by

Irshad Anwar

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