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From a variable point on the tangent at ...

From a variable point on the tangent at the vertex of a parabola `y^2=4a x ,` a perpendicular is drawn to its chord of contact. Show that these variable perpendicular lines pass through a fixed point on the axis of the parabola.

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Equation of chord of contact to parabola `y^(2)=4ax` w.r.t.
point `P(0,lamda)` is
`lamday=2ax`
`or2ax-lamday=0` (1)
Equation of line perpendicular to this line is `lamdax+2ay=c`.
If is passes through `p(0,lamda)` then `c=2alamda`.
Therefore, equation of line perpendicular to (1) through point P is
`lamdax+2ay=2alamda`
`or2ay+lamda(x-2a)=0`
This equation represents the equation of family of straight lines concurrent at point of intersection of lines y=0andx-2a=0.
So, lines are concurrent at (2a,0).
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Knowledge Check

  • The equation of the tangent to the parabola y^(2) = 8x which is perpendicular to the line x-3y+8=0 is -

    A
    `3x + y + 2 =0`
    B
    `3x - y- 1 =0`
    C
    `9x - 3y +2 =0`
    D
    `9x + 3y +2 =0`

  • The angle between the tangents drawn from the point (1, 4) to the parabola y^(2)=4x is -

    A
    `(pi)/(2)`
    B
    `(pi)/(6)`
    C
    `(pi)/(4)`
    D
    `(pi)/(3)`

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