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Point Form of Tangent: Hyperbola

Through the v...

Question

Through the vertex 'O' of the parabola $y^{2} = 4 a x$ , variable chords OA and OB are drawn at right angles. If the variable chord AB intersects the axis of $x$ at R, then distances OR :

varies with different positions of A and B

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Solution

The correct option is A varies with different positions of A and B
REF.Image
Let 'A' & 'B' be two points on the
parabola $y^{2} = 4 a x$
Since, $U A ⊥ U B$
$∴ (s l o p e o f U A) \times (s l o p e o f O B) = - 1$
$\frac{2 a l_{1}}{a l_{1}^{2}} \times \frac{2 a t_{2}}{a t_{2}^{2}} = - 1$
$\Rightarrow t_{1} t_{2} = - 4$ ___ (i)
Equation of 'AB' is given by
$y - 2 a t_{1} = \frac{2 a t_{2} - 2 a t_{1}}{a t_{2}^{2} - a t_{1}^{2}} (x - a t_{1}^{2})$
In x-axis $y = 0$
$0 - 2 a t_{1} = \frac{2 a (t_{2} - t_{1}) (x - a t_{1}^{2})}{a (t_{2} - t_{1}) (t_{2} + t_{1})}$
$\Rightarrow - 2 a t_{1} (t_{1} + t_{2}) = 2 (x - a t_{1}^{2})$
$\Rightarrow - a (t_{1}^{2} + t_{1} t_{2}) = x - a t_{1}^{2}$
$\Rightarrow - a t_{1}^{2} - a (- y) = x - a t_{1}^{2}$
$\Rightarrow x = 4 a$
Hence, R is $(4 a, U)$
$∴ U R = 4 a =$ Length of the lotus rectum

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