1.

A tangent is drawn at any point P(t) on the parabola y^(2)=8x and on it is takes a point Q(alpha,beta) from which a pair of tangent QA and OB are drawn to the circle x^(2)+y^(2)=8. Using this information, answer the following questions : The locus of the point of concurrecy of the chord of contact AB of the circle x^(2)+y^(2)=4 is

Answer»

`y^(2)-2x=0`
`y^(2)-x^(2)=4`
`y^(2)+4x=0`
`y^(2)-2x^(2)=4`

Solution :
(3)
The equation of the tangent at point P of the PARABOLA `y^(2)=8x` is
`yt=x+2T^(2)`(1)
The equation of the chord of contact of the circle `x^(2)+y^(2)=8` w.r.t. `Q(alpha,beta)` is
`xalpha+ybeta=8`(2)
`Q(alpha.beta)` lies on (1). Hence,
`betat=alpha+2t^(2)`(3)
`:.xalpha+y((alpha)/(t)+2t)-8=0`[From (2) and (3)]
`OR2(ty-4)+alpha(x+(y)/(t))=0`
For point of concurrency,
`x=-(y)/(t)andy=(4)/(t)`
Therefore, the LOCUS is `y^(2)+4x=0`


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