Locus and Parabola Questions (1 Viewer)

[allGenreGamer]

I can't do the below questions! please enlighten me:

Question 1: a) S is the focus (0, a)
P (2ap, ap^2)
Q (2aq, aq ^ 2)
I've proven PS+SQ = a(p^2 + q^2 + 2) as asked. The next questions reads: "Hence if PQ is a focal chord, show that the length of PQ = a(p +1/p)^2". I know i need to manipulate with PS+SQ = a(p^2 + q^2 + 2) but I fail to get PQ = a(p +1/p)^2! Please help.

b) If PQ is parallel to x-axis (latus rectum) prove PQ = 4a. This one I have no idea...

Question 2: Tangents are drawn t a parabola x^2 = 4y from an external point A(x1, y1), touching parabola at P and Q. Prove that th mid-point of PQ is (x1, 1/2x1^2 - y1).

Question 3: If PQ is focal chord of parabola x^2=4ay; QR is the tangent at Q and RP is parallel to the axis of the parabola, prove that the locus of R has the equation x^2(y+2a) = -4a^3

I've worked for hours and couldn't figure these out, fellow 3u students should know how this feels. Please help me

 

[George W. Bush]

Ah, now maths was something I learnt well at Yale.

1) PQ is a focal chord so pq = -1 i.e. p^2 = 1/q^2
Sub that into PS+SQ = a(p^2 + q^2 + 2) as obviously PS+SQ = PQ.
a(p^2 + 1/p^2 + 2) = a(p + 1/p)(p+ 1/p) = a(p + 1/p)^2

b) If PQ is parallel to the x-axis then ap^2 = a (as PQ is a focal chord) i.e. p = +- 1 = -q
Therefore length of PQ = 2a + 2a = 4a units.

Others later when I find some paper.

 

[mazza_728]

i cant do any questions wit the parabola! in class we've done 2 exercises and ive only been able to do one question by myself, and tht was the very first one.

 

[George W. Bush]

Q2: We know the midpoint of PQ is (p+q, 1/2(p^2 + q^2)

Now at the external point (x1, y1) the tangents will intercept so you solve the two tangent equations simultaneously
you'll get x1 = p+q, sub that into either tanget equation and manipulate it and you'll get the y co-ordinate.

q3: PQ is a focal chord so pq=-1, RP is parallel to the axis so the point R has the same x co-ordinate as the point P, use this to find x and y co-ordinates and then eliminate the parameter.