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

Asked by  | 23rd Feb, 2013, 03:04: PM

Expert Answer:

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.

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Answered by  | 28th Nov, 2017, 02:25: PM