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How to find the no of common tangents to the circles x^2 + y^2 = 4 , x^2+y^2 -6x-8y =24

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Find the distance between centers, (0,0) and (3,4) which is 5 units. Radius of first is 2 and second is 7. As C1C2=R-r, this implies circles are touching internally. So, only one common tangent is possible.
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Tutor

Find the distance between the centres of the two circles. Then compare it with the sum of the radii of them. You will come to know if they are touching or intersecting. Depending on the case you will be able to find out the number of common tangents.
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Tutor

Distance between centres = Sum of radii Three tangents. Two direct common tangent and one indirect common tangent. Distance between centres Sum of radii Two tangents. Two direct common tangents. Distance between centres > Sum of radii Four tangents. Two direct common tangent and two indirect... read more

Distance between centres = Sum of radii  Three tangents. Two direct common tangent and one indirect common tangent.

Distance between centres  Sum of radii  Two tangents. Two direct common tangents. 

Distance between centres > Sum of radii  Four tangents. Two direct common tangent and two indirect common tangents.

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Teacher, Mathematician, Tutor

Check centers and raddu of both circles and hence 1.) if they intersect then two common tangents 2.) If they touch then 3 common tangents 3.) If they do not touch then 4 common tangents
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Tutor

Centre of first Circle is A(0,0) and second centre is B(3,4). So distance between centres is 5. Radius of first circle=x=2 and radius of second y=7 So AB=|x-y|. So circles touch internally. Hence only one common tangent.
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R1 =2 R2 =5 R2-R1=3>0 So there is only one tangent is possible
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Tutor

These 2 cicle touch each other internally. You can find out the touching in the ratio of 2:7 externally (0,0) and (3,4) That point is point of contact. From that point T=0 , for any circle will give equation of tangent
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Mechanical Engineering Postgraduate from IIT

Let R and r be the radii of the two circles. Find out the distance d between the centres of the circles. Then, If d< abs(R-r), no common tangents If d = abs(R-r), one common tangent If abs(R-r) ≤ d ≤ max(R,r), two common tangents If d = max(R,r), three tangents If d > max(R,r), four... read more

Let R and r be the radii of the two circles. Find out the distance d between the centres of the circles. Then,

If d< abs(R-r), no common tangents

If d = abs(R-r), one common tangent

If abs(R-r) ≤ d ≤  max(R,r), two common tangents

If d = max(R,r), three tangents

If d > max(R,r), four tangents.

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Mathematics Tutor

Here only one common tangent can be drawn as the first circle is completely inside the second circle and touching internally since distance b/w their centres is equal to difference of their radii.
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Quicker Maths

First of all calculate the distance between centers of circle C (0,0) and C' (3,4) which is 5 units.By comparing general equation of circle, Radius of first circle is 2 unit and that of second is 7unit. As CC'=R-r, it means circles are touching internally. Hence, we can say that only one common tangent... read more

First of all calculate the distance between centers of circle C (0,0) and C' (3,4) which is 5 units.By comparing general equation of circle, Radius of first circle is 2 unit and that of second is 7unit. As CC'=R-r, it means circles are touching internally. Hence, we can say that only one common tangent is possible. 

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