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Question

A circle passes through the point (3,4) and cuts the circle x2+y2=a2 orthogonally; the locus of its centre is a straight line. If the distance of this straight line from the origin is 25, then a2=

A
250
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B
225
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C
100
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D
25
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Solution

The correct option is A 225
Note: Limiting Point of system of co-axial circles are the centers of the point circles belonging to the family.
As we know , every circle passing through limiting points of a coaxial system is orthogonal to all circles of the system.
So, any circle passing through a point and orthogonal to a given circle, will have its center on the radical axis of that point circle and that given circle. (or of the coaxial system)
So, let there be a circle with radius 0 and passes through (3,4). Equation of this circle S1:(x3)2+(y4)2=02
Given circle S2:x2+y2a2=0
Equation of Radical Axis :S1S2=0
6x8y+25+a2=0
This is the equation of a radical axis of a co-axial system of circles of which (3,4) is a limiting point. Hence, this will be the required locus.
Distance of (0,0) from this Radical Axis is 25. So,
25+a236+64=25
25+a2=250
a2=225

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