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Tangents are drawn to the circle x square plus y square equal to nine from a point on the hyperbola x square by nine minus y square by four equal to one.Find the locus of the mid point of the chord of contact. Please draw the figure.

Tangents are drawn to the circle x square plus y square equal to nine from a point on the hyperbola x square by nine minus y square by four equal to one.Find the locus of the mid point of the chord of contact. Please draw the figure.

Grade:11

1 Answers

Ashwin Muralidharan IIT Madras
290 Points
9 years ago

Hi Mayur,

 

Consider the point P(3sect,2tant) be any point on the hyperbola. (Refer the diagram, you'd asked for):

 

 

 

So the equation of the chord of contact from this point is (T=0)... which is

x(3sect) + y(2tant) = 9 ----------(1)

 

Now the equation of the chord of contact, in terms of it's midpoint M(h.k) is (T=S1) which is,

x(h) + y(k) = h2 + k2 ----------(2)

 

Both (1) and (2) represent the same straight line,

So 3sect/h = 2tant/k = 9/(h2 + k2)

to find the relation interms of h,k try to eliminate t from the above relation.

So, sect = 3h/(h2 + k2) and tant = 9k/2(h2 + k2).

And we know, sec2t - tan2t = 1,

So 9h2 - (9k/2)2 = (h2 + k2)2.

 

And hence the locus is 9x2 - (9y/2)2 = (x2+y2)2

 

Hope that helps.

Best Regards,

Ashwin (IIT Madras).

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