Question
Sat February 23, 2013 By:

if from any point on the common chord of two intersecting circles,tangents be drawn to the circles,prove that they are equal.

Expert Reply
Sat February 23, 2013
begin mathsize 16px style Let space the space two space circles space intersect space at space points space straight M space and space straight N. space MN space is space the space chord.
Suppose space straight O space is space straight a space point space on space the space common space chord space and space OP space and space OQ space be space the space tangents
drawn space from space straight A space to space the space circle.
OP space is space the space tangent space and space OMN space is space straight a space secant.
According space to space the space theorem space which space says space that space if space PT space is space straight a space tangent
to space the space circle space from space an space external space point space straight P space and space straight a space secant space to space the space circle
through space straight P space intersects space the space circle space at space points space straight A space and space straight B comma space then
PT squared equals PA cross times PB
Therefore comma space OP squared equals OM cross times ON.............. left parenthesis 1 right parenthesis
Similarly comma space OQ space is space the space tangent space and space OMN space is space straight a space secant space for space that space also.
T herefore comma space OQ squared equals OM cross times ON................ left parenthesis 2 right parenthesis
Comparing space left parenthesis 1 right parenthesis space and space left parenthesis 2 right parenthesis
OP squared equals space OQ squared rightwards double arrow OP equals OQ
Hence comma space the space length space of space the space 2 space tangents space is space equal.

end style
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