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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

Jitender Singh IIT Delhi
7 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
Soham Chitnis
21 Points
2 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