another parametrics question (1 Viewer)

[yugi]

two points P(6p,3p^2) and Q(6q,3q^2) lie on the parabola x^=12y
a) find the equation of the tangent at p
b) the tangents at p and q intersect at the point t, find the coords of the point t
c) if the angle between these tangents is 45 degrees prove that p-q=pq+1

i've done part a) and b) however i dont know how to do part c

for part a my answers were

a) y'=x/2a=6p/6=p
y-3p^2=p(x-6p)
.'.y=px-3p^2 ---- (1)
similarly the tangent at Q would be:
y=qx-3q^2 ---- (2)

b) solving (1) and (2) simultaneously
px-3p^2=qx-3q^2
x(p-q)=3(p+q)(p-q)
.'.x=3(p+q) ----- (3)
sub (3) into (1) to find y
.'.y=3pq

if someone could check my answers that would be great
thanks in advance

 

[shaon0]

two points P(6p,3p^2) and Q(6q,3q^2) lie on the parabola x^=12y
a) find the equation of the tangent at p
b) the tangents at p and q intersect at the point t, find the coords of the point t
c) if the angle between these tangents is 45 degrees prove that p-q=pq+1

i've done part a) and b) however i dont know how to do part c

for part a my answers were

a) y'=x/2a=6p/6=p
y-3p^2=p(x-6p)
.'.y=px-3p^2 ---- (1)
similarly the tangent at Q would be:
y=qx-3q^2 ---- (2)

b) solving (1) and (2) simultaneously
px-3p^2=qx-3q^2
x(p-q)=3(p+q)(p-q)
.'.x=3(p+q) ----- (3)
sub (3) into (1) to find y
.'.y=3pq

if someone could check my answers that would be great
thanks in advance

Yeah, think they're right. For c): Use (m1-m2)/(1+m1m2)=tan@ where m1=p and m2=q and @=45 deg.

 

[yugi]

oh i see, ill try that
thank you

 

[yugi]

btw would you by any chance know the formulas for internal/external divisions of an interval as well

 

[shaon0]

btw would you by any chance know the formulas for internal/external divisions of an interval as well

I did but i forgot most formulae from the HSC. It should be in any good textbook.