IIT-JEE 2009 · Paper 2 · Mathematics Q27

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MathematicsMultiple correct+4 / -1Question 27 of 57

The tangent $PT$ and the normal $PN$ to the parabola $y^2=4ax$ at a point $P$ on it meet its axis at points $T$ and $N$ , respectively. The locus of the centroid of the triangle $PTN$ is a parabola whose

vertex is

\left(\dfrac{2a}3,0\right)

directrix is

x=0

latus rectum is

\dfrac{2a}3

focus is

(a,0)

Answer and worked solution

Official answer: A, D

Points. For $P(at^2,2at)$ the tangent meets the axis at $T(-at^2,0)$ and the normal $y=-tx+2at+at^3$ at $N(2a+at^2,0)$ .

Centroid. $x=\dfrac{2a+at^2}3$ , $y=\dfrac{2at}3$ . Eliminating $t$ :
$y^2=\frac{4a}3\left(x-\frac{2a}3\right).$

Parabola data. Vertex $\left(\frac{2a}3,0\right)$ ; latus rectum $\frac{4a}3$ ; focus $\left(\frac{2a}3+\frac a3,0\right)=(a,0)$ ; directrix $x=\frac a3$ .

(A), (D)

Topics:Parabola Tangents and normals to conics Locus

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IIT-JEE 2009 · Paper 2, Mathematics question 27