Trigonometry> trigonometry-query...

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)

Aditya16 , 11 Years ago

Grade 12

1 Answers

Sumit Majumdar

Last Activity: 10 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,

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Trigonometry> trigonometry-query...

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)

Aditya16 , 11 Years ago

Sumit Majumdar

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