JEE Advanced 2014 · Paper 2 · Mathematics Q52

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MathematicsSingle correct+3 / -1Question 52 of 60

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Paragraph For Questions 51 and 52

Let $a$ , $r$ , $s$ , $t$ be nonzero real numbers. Let $P(at^2,2at)$ , $Q$ , $R(ar^2,2ar)$ and $S(as^2,2as)$ be distinct points on the parabola $y^2=4ax$ . Suppose that $PQ$ is the focal chord and lines $QR$ and $PK$ are parallel, where $K$ is the point $(2a,0)$ .

If $st=1$ , then the tangent at $P$ and the normal at $S$ to the parabola meet at a point whose ordinate is

\dfrac{(t^2+1)^2}{2t^3}

\dfrac{a(t^2+1)^2}{2t^3}

\dfrac{a(t^2+1)^2}{t^3}

\dfrac{a(t^2+2)^2}{t^3}

Answer and worked solution

Official answer: B

Lines. With $s=\frac1t$ :

tangent at P: $ty=x+at^2$ ;
normal at S: $y=-sx+2as+as^3$ .

Intersection. Substitute $x=ty-at^2$ into the normal and use $st=1$ :
$y=-y+at+\frac{2a}t+\frac a{t^3}\implies y=\frac a2\cdot\frac{t^4+2t^2+1}{t^3}=\frac{a(t^2+1)^2}{2t^3}.$

(B)

Topics:Tangents and normals to conics Parabola

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JEE Advanced 2014 · Paper 2, Mathematics question 52