Tangent line and normal on a parabola

[sooyong94]

Homework Statement

If the normal at P(ap^2 ,2ap) to the parabola y^2 = 4ax meets the curve again at Q(aq^2, 2aq), show that p^2 +pq+2=0

Homework Equations

Point-slope form

The Attempt at a Solution

I tried putting y=2aq and x=aq^2 but I can seem to simplify the whole thing other than dividing both sides by a

 

[ehild]

Homework Statement

If the normal at P(ap^2 ,2ap) to the parabola y^2 = 4ax meets the curve again at Q(aq^2, 2aq), show that p^2 +pq+2=0

Homework Equations

Point-slope form

The Attempt at a Solution

I tried putting y=2aq and x=aq^2 but I can seem to simplify the whole thing other than dividing both sides by a

The expression is also dividable by q-p.

 

[sooyong94]

Strangely enough I got this:

 

[ehild]

Strangely enough I got this:

There should be minus in front of the last term in the first equation. You made a mistake when copying.

 

[sooyong94]

Thanks - worked that out quickly.

 

[sooyong94]

Now I'm stuck on the second part:

Show that the equation of the locus of the point of intersection of the tangents at P and Q to the parabola is y^2(x+2a)+4a^3 =0. What does this mean? Does this mean that the tangents at P and Q meet at a point?

I managed to find the points of intersection of the two tangents (apq, a(p+q)), but I can't seem to continue at this point.

 

[ehild]

Show that the equation of the locus of the point of intersection of the tangents at P and Q to the parabola is y^2(x+2a)+4a^3 =0. What does this mean? Does this mean that the tangents at P and Q meet at a point?

I managed to find the points of intersection of the two tangents (apq, a(p+q)), but I can't seem to continue at this point.

You have to work with x and y. What are p and q now?

 

[sooyong94]

x=apq
y=a(p+q)

 

[ehild]

x=apq
y=a(p+q)

Have you copied the question correctly? These x and y values do not fulfill the equation given y^2(x+2a)+4a^3 =0.

 

[sooyong94]

I'm sorry - but I can't catch it.

 

[ehild]

I'm sorry - but I can't catch it.

I mean the problem might be wrong. The coordinates of the point of intersection do not fit to the given locus.