If the normals at three points P - Q - R of the parabola y 2-4 ax meet at b - c then the centriod of the triangle P QR lies on theA- axis of the parabolaB- directrix of the parabolaC- tangent at the vertexD- director circle

The correct option is A axis of the parabolaEquation of the normal to the parabola y^2=4ax is nbsp; y=mx 2am am^3 it passes through (b,c) ∴ c=mb 2am am^3 am ...

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Standard XII

Mathematics

Equation of Tangent When a Point Is Given

If the normal...

Question

If the normals at three points $P, Q, R$ of the parabola $y^{2} = 4 a x$ meet at $(b, c)$ then the centriod of the triangle $P Q R$ lies on the

axis of the parabola

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directrix of the parabola

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tangent at the vertex

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director circle

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Solution

The correct option is A axis of the parabola
Equation of the normal to the parabola $y^{2} = 4 a x$ is
$y = m x - 2 a m - a m^{3}$ it passes through $(b, c)$
$∴ c = m b - 2 a m - a m^{3} a m^{3} + (2 a - b) m + c = 0$
Let the roots of the equation are $m_{1}, m_{2}, m_{3}$
Then $m_{1} + m_{2} + m_{3} = 0 m_{1} m_{2} + m_{3} m_{2} + m_{1} m_{3} = \frac{2 a - b}{a} m_{1} m_{2} m_{3} = - \frac{c}{a}$
Feet of the normals are
$P \equiv (a m_{1}^{2}, - 2 a m_{1})$
$Q \equiv (a m_{2}^{2}, - 2 a m_{2})$
$R \equiv (a m_{3}^{2}, - 2 a m_{3})$
$y$ coordinate of the centriod of the triangle $P Q R$ is
$= \frac{- 2 a (m_{1} + m_{2} + m_{3})}{3} = 0$

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If the normals at three points P,Q,R of the parabola y2=4ax meet at (b,c) then the centriod of the triangle PQR lies on the

If the normals at three points $P, Q, R$ of the parabola $y^{2} = 4 a x$ meet at $(b, c)$ then the centriod of the triangle $P Q R$ lies on the