Locus Questions! Yipee! (1 Viewer)

[S1M0]

We rushed through this topic at an alarming speed, and i didn't pay attention..

Now i'm suffering with exams due tomorrow. So fulfill your BoS duties by easing the burden and helping me with a few questions. The straighforward stuff is no problem, but its when it gets to the tricky stuff that screws me over.

Anyway here goes:

1. The tangent with equation 2x - y - 4 = 0 touches the parabola x^2 = 4y at A. Find the co-ordinates of A


2. The focal chord that cuts the parabola x^2 = -6y at (6, -6) cuts the parabola again at X. Find the co-ordinates of X.

3. Find the co-ordinates of the endpoints of the latus rectum of the parabola x^2 = -8y. What is the length of the latus rectum.

Thankyou my fellow BoSers. Be swift. Swift like the wind.

 

[S1M0]

Bump.

Anyone....?????????????

 

[S1M0]

2nd bump.

I need to emphasise the urgency of these questions...

 

[Marzaa]

for question 1

2x - y - 4 = 0
y= 2x -4 (1)
y=x^2 / 4 (2)

substituting 1 into 2

2x - 4 = x^2 /4
therefore, x^2 = 8x -16
x^2 - 8x + 16 = 0
(x - 4)(x - 4) = 0
x = 4

therefore, y = 2(4) -4
= 4

hence A (4,4 )

i think,
soz dont hav time to help with other questions atm, im sure other helpful pplz will

 

[jb_nc]

For 1) set y = x²/4 then sub that into 2x - y - 4. You should get A = (4, f(4))

I can't really recall locus that well, BUT I had a go with them:

(This may be wrong) For 2) you should work out the focus point of the parabola using the formula x² = 4py then use (0, p) with (6, -6) to work out an the gradient and then an equation A for the line which passes through those points. Then use A to solve x² = -6y simultaneously to find point X. One of the points should be x = 6 and the other should be a member of R which is the point you are seeking to find.

3) (I found this via googling because I think it explains the q pretty well)The formula of a vertical parabola in vertex form with vertex (h,k) is: 4p(y - k) = (x - h)²

The axis of symmetry is the vertical line x = h. The directed distance from the vertex to the focus is p. The focus is therefore (h, k + p). The latus rectum is perpendicular to the line of symmetry. It runs from one side of the parabola, through the focus to the other side. It's length is 4p--a distance of 2p in either direction from the focus. The endpoints of the latus rectum are therefore:

(h - 2p, k + p) and (h + 2p, k + p)