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The centres of a set of circles each of radius 3 lie on the circle x2+y2=25. The locus of any point in the set is

The centres of a set of circles each of radius 3 lie on the circle x2+y2=25. The locus of any point in the set is

Grade:12

2 Answers

Arun
25750 Points
6 years ago
Dear student
 
Let C be the set of center points described by that equation x^2 + y^2 = 25. C is a circle of radius 5 centered at the origin. 
Let S be the set of radius 3 circles with centers in C. 
A point P is therefore on some circle in S if and only if it is exactly 3 units from some point in C. 
A little sketching of that should convince you that the distance of P from the origin must be at least 5-3 = 2 and at most 5+3 = 8; and that any such point is in fact 3 away from some point on C and therefore part of some circle in S. 
So, the points in you locus are the points (x,y) where: 
2 ≤ √(x² + y²) ≤ 8 
4 ≤ x² + y² ≤ 64 ... written without square roots
 
Regards
Arun (askIITians forum expert)
VIKASH RAVI
13 Points
2 years ago
Let C lie on x 
2
 +y 
2
 =25
 
Let S be the set of circles with radius 3 and center (h,k)
 
Any point P(h,k) on S will be 3 units from center C.
 
Distance from origin will be at least 5–3=2 and at most 5+3=8
 
⟹2≤ 
h 
2
 +k 
2
 
​
 ≤8
 
4≤h 
2
 +k 
2
 ≤64
 
Locus is
 
4≤x 
2
 +y 
2
 ≤64

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