If p and q are the lengths of the perpendiculars from the origin to the lines x cos θ - y sin θ = k cos 2θ and x sec θ + y cosec θ = k , respectively, prove that p^2 + 4q^2 = k^2.

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- September 10th 2008, 07:15 AMfardeen_gen[SOLVED] Straight lines(coordinate geometry)?
If p and q are the lengths of the perpendiculars from the origin to the lines x cos θ - y sin θ = k cos 2θ and x sec θ + y cosec θ = k , respectively, prove that p^2 + 4q^2 = k^2.

- September 10th 2008, 09:48 AMLaurent
The equation of the line can be written , where (and is the point we are describing). Then is a vector orthogonal to the line (if , then ), hence the projection of on is the point such that and , i.e. , so and , and the distance between the origin and the line is . This is a useful result to remember.

Using this fact, the exercise is just the verification of a simple trigonometric formula. It is even easier if you rewrite the second equation as (multiplying both sides by ), so that for both lines, so you have to check that . I guess you know how to do this.

Laurent. - September 10th 2008, 09:59 AMPlato
Here is a second way.

Using the standard distance formula we get:

and .

From that it follows .

Now you finish. - September 10th 2008, 11:16 AMLaurent
- September 10th 2008, 11:49 AMPlato