parametrics locus halp (1 Viewer)

[sssona09]

pls help
2. The tangent at a point P(2ap,ap^2) on the parabola x^2=4ay meets the x-axis at A and the y-axis at B. Find and describe the locus of the midpoint M of AB.

3. Consider the parabola x^2=4ay. If the normal at P(2p,p^2) and Q(2q,q^2) intersect at right angles at R, show that the locus of R is the parabola y=x^3+3

I tried but my answer for 3 was invalid :/

thank you all!!

 

[same1111]

pls help
2. The tangent at a point P(2ap,ap^2) on the parabola x^2=4ay meets the x-axis at A and the y-axis at B. Find and describe the locus of the midpoint M of AB.

3. Consider the parabola x^2=4ay. If the normal at P(2p,p^2) and Q(2q,q^2) intersect at right angles at R, show that the locus of R is the parabola y=x^3+3

I tried but my answer for 3 was invalid :/

thank you all!!

y=x^3+3 isnt a parabola btw. I also got a different answer for 3 so maybe the question is wrong.

 

[sssona09]

y=x^3+3 isnt a parabola btw. I also got a different answer for 3 so maybe the question is wrong.

y=x^2 +3 for q3 oops

wait can u post ur working out please?

 

[same1111]

https://ibb.co/gnB7fG

 

[sssona09]

https://ibb.co/gnB7fG

wait how is the y coordinate of R (the first line of working) a(p^2+q^2+pq+2)

like where is the 2 from

I thought its not there.. ?

 

[same1111]

wait how is the y coordinate of R (the first line of working) a(p^2+q^2+pq+2)

like where is the 2 from

I thought its not there.. ?

its the intersection of the normals.. If you solve the two equations for the normal of q and p you get that for the y coordinate.

 

[sssona09]

its the intersection of the normals.. If you solve the two equations for the normal of q and p you get that for the y coordinate.

ohh!! lol
thank YOU SO MUCH FOR THE WORKING OUT!!! MUCH LUb