Parametrics Question (1 Viewer)

[YashC3]

I found this question in the Purple Maths In Focus Book i.e. Maths Ext. Book 2 pg 396 qu. 2

Find the equations of the tangents to the curve x squared = 8y at the points P(4p, 2p squared) and Q (4q, 2q squared). Find the equation of the locus of their points of intersection if PQ is a focal chord.

Any help will be appreciated. Cheers

 

[shaon0]

I found this question in the Purple Maths In Focus Book i.e. Maths Ext. Book 2 pg 396 qu. 2

Find the equations of the tangents to the curve x squared = 8y at the points P(4p, 2p squared) and Q (4q, 2q squared). Find the equation of the locus of their points of intersection if PQ is a focal chord.

Any help will be appreciated. Cheers

y=x^2/8
y'=x/4
At P:
y'=p

y-2p^2=p(x-4p)
y=px-2p^2
Similarly, equ of tangent at Q:
y=qx-2q^2

px-2p^2=qx-2q^2
x(p-q)=2(p+q)(p-q)
x=2(p+q) => y=2pq

For focal chord: pq=-1 as tangents meet at 90 degrees
y=-2

 

[YashC3]

Cheers for the help

 

[YashC3]

I worked through the rest of the exercise but I got stuck on question 3.

Find the equation of the locus of the point R that is the intersection of the normals at P(2p, p^2) and Q (2q, q^2) on the parabola x^2 = 4y, given that pq = -4

I know exactly what to do, but I was having trouble finding the pt. of intersection of the two normals.

I got two equations: y= -x/p + p^2 + 2

and

y= -x/q + q^2 + 2

Anyone else get the same equations? And how do I find the pt of intersection from there.

Thanks alot

 

[frenzal_dude]

y=x^2/8
y'=x/4
At P:
y'=p

y-2p^2=p(x-4p)
y=px-2p^2
Similarly, equ of tangent at Q:
y=qx-2q^2

px-2p^2=qx-2q^2
x(p-q)=2(p+q)(p-q)
x=2(p+q) => y=2pq

For focal chord: pq=-1 as tangents meet at 90 degrees
y=-2

Wouldn't the gradient at P be (p^2)/(2p) =(1/2)p ?