From an arbitrary point P on the circle x squared plus y squared space equals space 4 comma space tan g e n t s space a r e space d r a w n space t o space t h e space c i r c l e space x squared plus y squared space equals space 1 which meet again at points A & B. The locus of point of intersection of tangents drawn to the former circle at
A & B is
 
 
options :
1) a line                           2) a circle of radius 3 unit 
3) An ellipse                     4) a conic with center at the origin.
 
 
 

Asked by Sunil Soni | 10th Aug, 2016, 05:19: PM

Expert Answer:

 
 
 
 
 
 
Let space straight O space be space the space centre space of space the space circle space straight x squared plus straight y squared equals 4 space space and space straight O apostrophe space be space the space centre space of space the space circle space straight x squared plus straight y squared equals 1.
Let space straight P space be space an space arbitrary space point space on space the space circle space with space centre space straight O.

Draw space tangents space from space point space straight P space on space the space circle space with space centre space straight O apostrophe space touching space at space space straight A space and space straight B. space
We space know space that space tangents space drawn space from space an space external space point space are space equal space in space length.
rightwards double arrow PA space equals space PB

Again space draw space tangents space from space points space straight A space and space straight B space on space the space circle space with space centre space straight O space touching space at space space straight C space and space straight D.
T angents space drawn space from space an space external space point space are space equal space inlength.
rightwards double arrow AC space equals space AD space equals space BC space equals space BD

Let space straight M space be space the space point space of space intersection space of space tangents space PA space and space BC.
Let space straight N space be space the space point space of space intersection space of space tangents space PB space and space AD.

rightwards double arrow straight M space and space straight N space are space equidistant space from space the space points space straight A space and space straight B.
Also space straight M space and space straight N space are space equidistant space from space the space points space straight A space and space straight B.

So space we space can space draw space straight a space vertical space line space psssing space through space them.

Hence space the space locus space of space point space of space intersection space of space tangents space drawn space to space the space former space circle space at space straight A space & space straight B space is space straight a space straight space line.
 
 
 
 
 

Answered by Vijaykumar Wani | 12th Aug, 2016, 02:30: PM

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