The normal at a point on the parabola y^2=4x | vedclass

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(D) The equation of the parabola is $y^2=4ax$,where $a=1$. The normals to the parabola $y^2=4ax$ passing through a point $(h,k)$ are given by the cubi

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$(2,0)$

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(D) The equation of the parabola is $y^2=4ax$,where $a=1$. The normals to the parabola $y^2=4ax$ passing through a point $(h,k)$ are given by the cubic equation $my^3 + (2a-h)m^2 + k^2m - k = 0$. For the point $(5,0)$,we have $h=5$ and $k=0$. The equation becomes $m(y^2 + (2-5)) = 0$,which simplifies to $m(y^2-3)=0$. The feet of the normals $(x_i, y_i)$ for $i=1, 2, 3$ are the points on the parabola. The centroid $(G_x, G_y)$ of the triangle formed by the feet of the normals $(x_1, y_1), (x_2, y_2), (x_3, y_3)$ is given by $G_x = \frac{x_1+x_2+x_3}{3}$ and $G_y = \frac{y_1+y_2+y_3}{3}$. For a parabola $y^2=4ax$,the centroid of the triangle formed by the feet of the three normals passing through $(h,k)$ is $\left(\frac{2}{3}(h-2a), 0\right)$. Substituting $h=5$ and $a=1$,we get the centroid as $\left(\frac{2}{3}(5-2(1)), 0\right) = \left(\frac{2}{3}(3), 0\right) = (2,0)$.

The normal at a point on the parabola $y^2=4x$ passes through $(5,0)$. If there are two more normals to this parabola which pass through $(5,0)$,the centroid of the triangle formed by the feet of these three normals is

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