given that S and S' are the focii of hyp with eq/n x^2/a^2 - y^2/b^2 = 1 show that SPand S'P are eqaully inclined to the tangent at any point P on the curve
prove that the triangle formed formed by the asymptotes of the curve with eq/n x^2 - 2y^2 = 4 and any tangent to the curve is of constant area
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- Thread Starter
- 03-02-2005 13:19
- 16-02-2005 12:09
Going to bring this back to the top, as also would like help on this question
thankyou , Ilana
- 16-02-2005 12:54
For the 1st question, you can use the reflection propety of a hyperbola.
Can you just quote it? Otherwise you may have to go through a lot of trig.
- 16-02-2005 13:18
Let P be a point on the curve with coords (x0, y0). Then the equation of the tangent at P is:
x(x0) - 2y(y0) = 4
And the equation of the asymptotes is:
x^2 = 2y^2
Let the asymptotes intersect the tangent at point P, then:
(x^2)/2 = y^2
x(x0) - 4 = 2y(y0)
x(x0) - 4 = 2(sqrt[x^2/2])(y0)
[x(x0) - 4]^2 = 4x^2(y0)/2 = 2x^2(y0)^2
2x^2(y0)^2 - x^2(x0)^2 + 8x(x0) - 16 = 0
x^2[2(y0)^2 - (x0)^2] + 8x(x0) - 16 =0
If you compare 2(y0)^2 - (x0)^2 to the equation of the curve, you'll see that 2(y0)^2 - (x0)^2=-4. So:
-4x^2 + 8(x0)x - 16 =0
x^2 - 2(x0)x + 4 = 0
This equation has roots a & b, which are the coordinates of the point of intersection between the tangent and the asymptotes. a*b=4, so the product of the x-coordinates is always constant, so the lengths of the lines connecting the origin to the vertices of the triangle is constant. So the triangle is of constant area.
Hopefully my reasoning is too off.
- 17-02-2005 02:49
If you can't use the reflection property, here's a solution to the problem.