# Tangent line and normal on a parabola

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1. Aug 14, 2016

### sooyong94

1. The problem statement, all variables and given/known data
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

2. Relevant equations
Point-slope form

3. 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

2. Aug 14, 2016

### ehild

The expression is also dividable by q-p.

3. Aug 14, 2016

### sooyong94

Strangely enough I got this:

4. Aug 14, 2016

### ehild

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

5. Aug 14, 2016

### sooyong94

Thanks - worked that out quickly.

6. Aug 15, 2016

### 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.

7. Aug 15, 2016

### ehild

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

8. Aug 15, 2016

### sooyong94

x=apq
y=a(p+q)

9. Aug 15, 2016

### ehild

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

10. Aug 15, 2016

### sooyong94

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

11. Aug 16, 2016

### ehild

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