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If tangents are drawn to y^2=4a x from a...

If tangents are drawn to `y^2=4a x` from any point `P` on the parabola `y^2=a(x+b),` then show that the normals drawn at their point for contact meet on a fixed line.

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Let `P-=(at_(1)t_(2),a(t_(1)+t_(2))),P` must satisfy `y^(2)=a(x+b)`. Hence,
`a^(2)(t_(1)+t_(2))^(2)=a{(at_(1)t_(2))+}`
`or" "a(t_(1)^(2)+t_(2)^(2)+t_(1)t_(2))=b`
Now, the coordinates of the point of intersection of normals at `t_(1)andt_(2)` are, respectively,
`h=a(t_(1)^(2)+_(2)^(2)+t_(1)t_(2)+2)` (1)
`and" "k=-at_(1)t_(2)(t_(1)+t_(2))` (2)
From (1)
h=b+2a
or x=b+2a
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