To find the equation of two circles with centers A(1, 1) and B(9, 7) which have an equal radius and touch each other externally and also the equation of common tangents to these circles.

Question

To find the equation of two circles with centers A(1, 1) and B(9, 7) which have an equal radius and touch each other externally and also the equation of common tangents to these circles.

Arham Warner · Accepted Answer

Step 1  Consider the diagram of the given situation:    Here  AP is the radius of the circle with center A  BP is the radius of the circle with center B  Given AP=BP=r  AB=(9−1)2+(7−1)2  AB=64+36  AB=100  AP+BP=10  r+r=10  2r=10  r=5  Therefore the radius of the circles is 5  Equation of circle with center C(h, k) and radius r is  (x−h)2+(y−k)2=r2  Here  Equation of circle with center A(1, 1) and radius 5 is  (x−1)2+(y−1)2=25  Equation of circle with center A(9, 7) and radius 5 is  (x−9)2+(y−7)2=25  Step 2  Here point P divides the line segment into 1:1 ratio, So the coordinate of point P is  P=(1+92, 7+12)=(102, 82)=(5, 4)  Line AP is perpendicular to tangent line, therefore  m1×m2=−1  4−15−1×m2=−1  m2=−43  Equation of tangent line is y=−43x+c and it passes through point P(5, 4) so  y=−43x+c  4=−43×5+c  c=4+203  c=323  Hence the required equation of common tangent is y=−43x+323

To find the equation of two circles with centers A (1,\ 1) and B (9,\ 7) which have an equal radius and touch each other externally.

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