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A quadrilateral is inscribed in a parabo...

A quadrilateral is inscribed in a parabola `y^2=4a x` and three of its sides pass through fixed points on the axis. Show that the fourth side also passes through a fixed point on the axis of the parabola.

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Consider quadrilateral joining point `A(t_(1)),B(t_(2)),C(t_(3))andD(t_(4))` is inscribed in the parabola `y^(2)=4ax`.
Equation of line AB is
`y-2at_(1)=(2)/(t_(1)+t^(2))(x-at_(1)^(2))`
`or2x-(t_(1)+t_(2))y+2at_(1)t_(2)=0`
Let AB pass through fixed point (c,0) on x-axis.
`:." "t_(1)t_(2)=-(c)/(a)` . . .(1)
Similarly, for chord BC,
`t^(2)t^(3)=-(d)/(a)` (2)
and for chord CD,
`t_(3)t_(4)=-(e)/(a)` (3)
Multiplying (1) and (3),
`t_(1)t_(2)t_(3)t_(4)=(ec)/(a^(2))`
`rArr" "t_(1)t_(4)=-(ec)/(ad)`
Thus, chord AD also passes through a fixed on x-axis.
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