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# If the tangent at point P of the curve, y^2=x^3 intersects the curve again at point Q and the straight lines OP,OQ make angles a,b withe x axis where O is origin, then tan(a)/tan(b) has a value-(1.) -1    (2). -2    (3). 2       (4).  2^(.5)

Sumit Majumdar IIT Delhi
7 years ago
Solution:
Let us assume the parametric coordinates at P and Q be:
$\left (t_{1}^{2}, t_{1}^{3} \right ), \left (t_{2}^{2}, t_{2}^{3} \right )$
Now finding the slope of the tangent at the point P, we get:
$\frac{dy}{dx}=\frac{3x^{2}}{2y}=\frac{3}{2t_{1}}$
Also the line joining P and Q would have a slope given by:
$\frac{t_{2}^{3}-t_{1}^{3}}{t_{2}^{2}-t_{1}^{2}}$
equating both the slopes would give:
$2t_{2}^{2}=t_{1}\left ( t_{2}+t_{1} \right )$
This can futher be simplified as:
$2\frac{t_{2}^{2}}{t_{1}^{2}}=\frac{t_{2}}{t_{1}}+1$
Also, we have:
$tan(a)=t_{1} tan(b)=t_{2}$
Using these we get the required result.
Thanks & Regards
Sumit Majumdar,