Have you done any work so far? Try typing up as much as you can.
So, okay I don't even know what the question means. I'm so confused now..
Consider the parabola , where and suppose the tangents at and intersect at point T.
Let be the focus of the parabola.
1) Find the coordinates of T. (assume that the tangents at P is ).
2) Show that . Suppose P and Q move on the parabola in such a way that
3) Show that T is constrained to move on a parabola
For (1), you have it correct. For (2), I am not sure what you are doing. You are supposed to show that . You can do that by using the distance formula. LaTeX is not working, so plugging in the information to the distance formula:
[tex]SP = \sqrt{ (2ap - 0)^2 + (ap^2 - a)^2 }[/tex]
Simplifying, you will get as you wanted. Similarly, [tex]SQ = a(q^2+1)[/tex].
Then, for (3), you suppose that [tex]SP+SQ=4a[/tex], then show that the point T will move along a parabola.
So, you get [tex]SP+SQ = a(p^2+1) + a(q^2+1) = a(p^2+q^2+2) = 4a[/tex]. Hence, [tex]p^2 + q^2 = 2[/tex].
From the formula for T you found in part (1), you know [tex]x = a(p+q)[/tex]. Squaring both sides, you get
[tex]x^2 = a^2(p+q)^2 = a^2(p^2+2pq+q^2) = a^2(p^2+q^2) + 2a(apq) = 2a^2 + 2ay[/tex]
Solving for y, you get: [tex]y = \dfrac{1}{2a}x^2 - a[/tex] which is the definition for a parabola as desired.
Thanks SlipEternal
But im kind of confused in the distance formula,
SP = \sqrt{ (2ap - 0)^2 + (ap^2 - a)^2 }
But why do you use 2ap in (2ap - 0)^2 instead of ap ? P's x coordinate is ap right?
No, it is 2ap. You had a typo. The point (ap,ap^2) is not a point on the parabola. Here is why:
4ay = x^2 is the formula for the parabola. Plug in ap for x and ap^2 for y: 4a(ap^2) = 4a^2p^2 does not equal a^2p^2. On the other hand, if you change x to 2ap, on the right hand side you get 4a^2p^2, which is the same as the left hand side.