Question

Let a, r, s, t be non zero real numbers. Let P(at²,2at), Q, R(ar²,2ar) and S(as²,2as) be distinct points on the parabola y² = 4ax. Suppose that PQ is the focal chord and lines QR and PK are parallel, where K is the point (2a, 0).

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Linked Question 1

The value of r is

$\frac{1}{t}$

$\frac{t^{2} + 1}{t}$

$\frac{t^{2} - 1}{t}$

$\frac{- 1}{t}$

m_QR = m_PK

$\frac{2 ar + \frac{2 a}{t}}{{ar}^{2} - \frac{a}{t^{2}}} = \frac{2 at - 0}{{at}^{2} - 2 a}$

$\Rightarrow t^{2} - 2 = t (r - \frac{1}{t})$ ;

$∴ r = - \frac{t^{2} - 1}{t}$

Linked Question 2

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

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

$\frac{{(t^{2} + 1)}^{2}}{2 t^{3}}$

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

$\frac{a {(t^{2} + 2)}^{2}}{t^{3}}$

Tangent at P is

yt = x + at² .....(1)

Normal at S (as², 2as) is y + sx + 2as + as³

Put $s = \frac{1}{t} \Rightarrow y + \frac{x}{t} = \frac{2 a}{t} + \frac{a}{t^{3}}$ .....(2)

from (1) and (2), eliminate x, we get

$y = \frac{a {(t^{2} + 1)}^{2}}{2 t^{3}} .$