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Consider a circle S with centre at the origin and radius 4. Four circles A, B, C and D each with radius unity and centres (–3, 0), (–1, 0), (1, 0) and (3, 0) respectively are drawn. A chord PQ of the circle Stouches the circle B and passes through the centre of the circle C. If the length of this chord can beexpressed as root x , find x

Consider a circle S with centre at the origin and radius 4. Four circles A, B, C and D each with radius unity and centres (–3, 0), (–1, 0), (1, 0) and (3, 0) respectively are drawn. A chord PQ of the circle Stouches the circle B and passes through the centre of the circle C. If the length of this chord can beexpressed as root x , find x

Grade:11

2 Answers

Jitender Singh IIT Delhi
askIITians Faculty 158 Points
9 years ago
Ans: 120
Sol:
PQ is a tangent to circle B.
Circle B:
x^{2} + y^{2} + 2x = 0
Equation of PQ
xx_{1} + yy_{1} + x + x_{1} = 0
It is passing through (1, 0)
x_{1} + x_{1} + 1 = 0
x_{1} = \frac{-1}{2}
Putting this value in circle B, we have
y_{1} = \frac{\pm (3)^{1/2}}{2}
Equation of PQ:
x + (3)^{1/2}y = 1
It is intersecting with circle S at point P & Q.
Circle S:
x^{2} + y^{2} = 16
After finding intersection of PQ with circle S, we have
P = (1-\frac{3(10)^{1/2}}{2}, \frac{(30)^{1/2}}{2})
Q = (1+\frac{3(10)^{1/2}}{2}, \frac{-(30)^{1/2}}{2})
PQ = ((3(10)^{1/2})^{2}+ ((30^{1/2})^{2})^{1/2}
x = 120
Cheers!
Thanks & Regards
Jitender Singh
IIT Delhi
askIITians Faculty
Soham Chitnis
21 Points
5 years ago
Wrong answer x=63 do not find co ordinates of p and q find the perpendicular distance from the Origin to the found line which will 1/2 
 Length of chord =2×(√(16-1/4)=√63 
x=63

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