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Question Number 96340 by Rio Michael last updated on 31/May/20

The equations of two circles S_{1} and S_{2} are given by

S_{1} : x^{2} + y^{2} + 2 x + 2 y + 1 = 0

S_{2} : x^{2} + y^{2} - 4 x + 2 y + 1 = 0 .

Show that S_{1} and S_{2} touch each other externally and obtain

the equation of the common tangent T at the point of contact .

Commented by PRITHWISH SEN 2 last updated on 31/May/20

The distance between the centers = The sum of

their radius

Now prove it . I know you can .

Commented by Rio Michael last updated on 31/May/20

so you mean C_{1} C_{2} = r_{1} + r_{2}

S_{1} : {(x + 1)}^{2} + {(y + 1)}^{2} = 1^{2} \Rightarrow C_{1} (- 1, - 1), r_{1} = 1

S_{2} : {(x - 2)}^{2} + (y + 1) = 2^{2} \Rightarrow C_{1} (2, - 1), r_{2} = 2

C_{1} C_{2} = \sqrt{3^{2}} = 3 units .

r_{1} + r_{2} = 1 + 2 = 3 units

\Rightarrow they touch each other externally .

sir whats the idea for the second part ?

Commented by PRITHWISH SEN 2 last updated on 31/May/20

Find the common point by solving the two

equations . And then get the slope of the line joining

that point with any one center .

Then keep in mind that the radius and the tangent

at the common point are ⊥ to each other .

So find the slope of the tangent . And then \dots . .

you know what to do .

Commented by Rio Michael last updated on 01/Jun/20

thank you sir .