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Q. conics (1 Viewer)

marsenal

cHeAp bOoKs
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Nov 12, 2002
Messages
273
From Patel's blue book:

1.Show that the line lx+my+n=0 touches the
(a)ellipse x<sup>2</sup>/a<sup>2</sup> + y<sup>2</sup>/b<sup>2</sup>=1, if a<sup>2</sup>l<sup>2</sup> + b<sup>2</sup>m<sup>2</sup>=n<sup>2</sup>
(b) hyperbola x<sup>2</sup>/a<sup>2</sup>-y<sup>2</sup>/b<sup>2</sup>=1, if a<sup>2</sup>l<sup>2</sup>-b<sup>2</sup>m<sup>2</sup>=n<sup>2</sup>

2.The tangent at P(x<sub>1</sub>,y<sub>1</sub>) on the hyperbola x<sup>2</sup>/a<sup>2</sup>-y<sup>2</sup>/b<sup>2</sup>=1, x<sub>1</sub> >1, intersects the directorix at Q. S is the focus (ae,0). Prove that PSQ is a right angle.

3. Find the equations of the four tangents common to the hyperbola x<sup>2</sup>-2y<sup>2</sup>=4 and the circle x<sup>2</sup>+y<sup>2</sup>=1. Find the points of contact of these tangents with the circle.

Thanx!
 

Affinity

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HSC
2003
Q 1 -> the 2 equations solved simultaneously gives a double root..

Q 2 -> find the gradients of PS SQ then... with that show they are perpendicular

Q 3 ->
derive tangent equation of circle at (cos(t),sin(t))
then use 1 -> (b) to find t.
 

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