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L O L ' and M O M ' are two chords of pa...

`L O L '` and `M O M '` are two chords of parabola `y^2=4a x` with vertex `A` passing through a point `O` on its axis. Prove that the radical axis of the circles described on `L L '` and `M M '` as diameters passes though the vertex of the parabola.

Text Solution

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Equation of `LOL':2x-(t_(1)+t_(2))y+2at_(1)t_(2)=0`
Since chord LOL' passes through (c,0), we have
`t_(1)t_(2)=-(c)/(a)`
Similarly, for chord MOM',
`t_(3)t_(4)=-(c)/(a)`
Now, the equation of the circle with LL' as diameter is
`(x-at_(1)^(2))(x-at_(2)^(2))+(y-2at_(1))(y-2at_(2))=0`
`or" "x^(2)+y^(2)-a(t_(1)^(2)+t_(2)^(2))x-2a(t_(1)+t_(2))y+c^(2)-4ac=0` (1)
`(Asa^(2)t_(1)^(2)t_(2)^(2)=c^(2)and4a^(2)t_(1)t_(2)=-4ac)`
Similarly, for the circle with MM' as diameter, the equation is
`x^(2)+y^(2)-a(t_(3)^(2)+t_(4)^(2))x-2a(t_(3)+t_(4))y+c^(2)-4ac=0` (2)
Radical axis of (1) and (2) is (1) - (2)
`or" "a(t_(1)^(2)+t_(2)^(2)-t_(3)^(2)-t_(4)^(2))x-2a(t_(1)+t_(2)-t_(3)-t_(4))y=0`
which passes through the origin (vertex of the parabola).
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