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Question

Let S1=x2+y2=9 and S2=(x2)2+y2=1. Then the locus of the Centre of a variable circle S which touches S1 internally and S2 externally always passes through the points:


A

12,±52

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B

2,±32

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C

1,±2

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D

(0,±3)

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Solution

The correct option is B

2,±32


Explanation for the correct option:

Finding the required points

Step 1: Finding the center and radius of the two circles

The given circle S1x2+y2=9

Therefore, C1(0,0)and r1=3 be the Centre and radius respectively.

Also, S2=(x2)2+y2=1

And also given C2(2,0) and , r2=1 be the Centre and radius respectively.

Considering the Centre of the variable circle be C3(h,k) and the radius be r.

Step 2: Illustrating the given information

JEE Main 2021 March 18 Shift 2 Section A Maths Question Paper with Solutions Q8

Step 3: Calculating the distance between the centre

C3C1=3r

Similarly,

C2C3=1+rC3C1+C2C3=4

Step 4: Using the concept of an ellipse

Therefore, the locus is an ellipse whose foci are C1&C2

And major axis is 2a=4and also 2ae=C1C2=2

e=12b2=4114=3[e=1-ba2]

Thus, the center of the ellipse is the midpoint of C1&C2is (1,0)

Now the equation of the ellipse

x1222+y232=1[x-x12a2+y-y12b2=1]

So now taking (x,y)=2,±32

21222+32232=114+34=11=1

Hence, it always passes through the point 2,±32.

Thus, option (B) is the correct answer.


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