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Two Point Form of a Line

Given the equ...

Question

Given the equations of two circles
$x^{2} + y^{2} = r^{2}$ and $x^{2} + y^{2} - 10 x + 16 = 0$ the value of r such that they intersect in real and distinct points is given by $2 < r < 8$

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Solution

we know that the two circles touch if
$C_{1} C_{2} = r_{1} + r_{2}$ [Externally]
$C_{1} C_{2} = r_{1} - r_{2}$ [Internally]
In case they intersect in real points , then
$C_{1} C_{2} < r_{1} + r_{2}$ and $C_{1} C_{2} > r_{1} - r_{2}$ ...(1)
$C_{1}$ = (0 , 0) , $C_{2}$ = (5 , 0) , $r_{1}$ = r , $r_{2}$ = 3
$∴$ 2 < r and 8 > r i.e. r < 8
Hence the required condition is $2 < r < 8$

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