Prove that the following circles are such that each of them touches the other two. Determine the points of contact . Show also that the three tangents concur at the point (-3,-3)
$S_{1} \equiv x^{2} + y^{2} + 4 y - 1 = 0$
$S_{2} \equiv x^{2} + y^{2} + 6 x + y + 8 = 0$
$S_{3} \equiv x^{2} + y^{2} - 4 x - 4 y - 37 = 0$

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Solution

Show that $C_{1} C_{2} = r_{1} + r_{2}$ or $r_{1} r_{2}$ .
The points of contact are to be found by ratio formula.Common tangents are given by
$S_{1} S_{2} = 0, S_{2} S_{3} = 0, S_{3} S_{1} = 0$
which can be shown to be concurrent at (- 3,- 3).

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Prove that the following circles are such that each of them touches the other two. Determine the points of contact . Show also that the three tangents concur at the point (-3,-3)S1≡x2+y2+4y−1=0S2≡x2+y2+6x+y+8=0S3≡x2+y2−4x−4y−37=0

Show that C1C2=r1+r2 or r1r2.The points of contact are to be found by ratio formula.Common tangents are given byS1S2=0,S2S3=0,S3S1=0which can be shown to be concurrent at (- 3,- 3).

Show that $C_{1} C_{2} = r_{1} + r_{2}$ or $r_{1} r_{2}$ .
The points of contact are to be found by ratio formula.Common tangents are given by
$S_{1} S_{2} = 0, S_{2} S_{3} = 0, S_{3} S_{1} = 0$
which can be shown to be concurrent at (- 3,- 3).