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Tangent is drawn at any point (p ,q) on ...

Tangent is drawn at any point `(p ,q)` on the parabola `y^2=4a xdot` Tangents are drawn from any point on this tangant to the circle `x^2+y^2=a^2` , such that the chords of contact pass through a fixed point `(r , s)` . Then `p ,q ,r` and`s` can hold the relation
(a) `r^2q=4p^2s` (b) `r q^2=4p s^2` (c)`r q^2=-4p s^2` (d) `r^2q=-4p^2s`

Text Solution

Verified by Experts

Equation of tangent to given parabola at point (p,q) on it is
qy=2a(x+p)
Let `(x_(1),y_(1))` be a point on this tangent. Then
`qy_(1)=2a(x_(1)+p)` (1)
Equation of chord of contact to the given circle w.r.t. point `(x_(1),y_(1))` is
`x x_(1)+yy_(1)=a^(2)` (2)
Given that this passes through point (r,s).
`:." "rx_(1)+sy_(1)=a^(2)`
From (1), putting the value of `y_(1)` in (2), we get
`x x_(1)+y(2a)/(q)(x_(1)+p)=a^(2)`
`:." "x_(1)(x+(2ay)/(q))+((2ap)/(q)y-a^(2))=0`
`rArr" "x_(1)(qx+2ay)+(2apy-a^(2)q)=0`
This is the variable line which passes through the point of intersection of lines `qx+2ay=0and2apy=a^(2)q`.
`rArr" "x=-(a^(2))/(p),y=(aq)/(2q)`
`:." "r=-(a^(2))/(p)ands=(aq)/(2p)`
`rArr" "4ps^(2)=-rq^(2)`
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