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The normal at a point on the parabola $y^2=4 x$ 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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The correct answer is:
$(2,0)$
We have, $y^2=4 x \quad \therefore \quad a=1$
We know that, the centroid of the triangle formed by the conormal points on a parabola lies on its axis and its coordinates are $\left(\frac{2}{3}(h-2 a), 0\right)$ where $(h, k)$ is the point through which all normals passes.
$\therefore \text { Coordinate of centroid }=\left(\frac{2}{3}(5-2 \times 1), 0\right)=(2,0)$
We know that, the centroid of the triangle formed by the conormal points on a parabola lies on its axis and its coordinates are $\left(\frac{2}{3}(h-2 a), 0\right)$ where $(h, k)$ is the point through which all normals passes.
$\therefore \text { Coordinate of centroid }=\left(\frac{2}{3}(5-2 \times 1), 0\right)=(2,0)$
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