Conics Question From Fitzpatrick. (1 Viewer)

[Eddyah]

Exercise 32a. Question 8

I'll try word it simply for those who dont have the book,

Point A is a moving point on the Y axis and point B is a moving point on the X-axis. The distance of AB is a constant of 6 units and point P is the point that divides AB by ratio of 1:2

a) Show several positions of A and B oneither sides of the axes and show the corresponding positions of P.

b) Show the locus of P is an elipse and find the equation.

I can easily show that a=2, b=4 therefore the ellipse has the equation x^2 /4 + y^2 /16 =1
But i can't 'prove' it algebraically.

 

[Eddyah]

wooo hooojust solved it.

P(b/3, 2a/3)


AB^2 = a^2 + b^2

x=b/3 y=2a/3

b=3x, a= 3y/2

Abrakadabra! we have our equation

 

[blakwidow]

:rofl:

 

[victorheaven]

let the moving point A be (0,a) and B(b,0)
since their distance is 6
therefore b^2+a^2=6 (equation 1)
P divides AB in the ratio of 1:2, so P has the coordinates of (b/3,2a/3) draw a graph if u cant understand
x=b/3 y=2a/3 ==> b=3x and a=3y/2 substitute these two into equation 1
u will get 12x^2+3y^2=8