Parametrics (1 Viewer)

[yeyjasminetime]

Hi if anyone has time could you please help me out? x

The points P(2ap. ap^2) and Q (2aq, aq^2) lie on the parabola D whose equation is x^2 = 4ay.

Suppose that A (intersection of the tangents to D at P and Q) lies on the line containing the latus rectum of D.
Show that pq=1

HOW CAN THE INTERECTION OF A TANGENT LIE ON THE LATUS RECTUM? D:

Please and thank you in advance

 

[Carrotsticks]

Hi if anyone has time could you please help me out? x

The points P(2ap. ap^2) and Q (2aq, aq^2) lie on the parabola D whose equation is x^2 = 4ay.

Suppose that A (intersection of the tangents to D at P and Q) lies on the line containing the latus rectum of D.
Show that pq=1

HOW CAN THE INTERECTION OF A TANGENT LIE ON THE LATUS RECTUM? D:

Please and thank you in advance

Lies on the line containing the latus rectum of D.

So not the latus rectum itself, but the horizontal line containing it (in this case, y=a).

 

[yeyjasminetime]

Lies on the line containing the latus rectum of D.

So not the latus rectum itself, but the horizontal line containing it (in this case, y=a).

Thank you!

 

[yeyjasminetime]

Hi Guys,

[An alternative proof of the reflection property] P is the variable point (x₁, y₁) on the parabola x² = 4ay, and S is the focus. The tangent at P meets the tangent at the vertex of the parabola at Q an it meets the axis of the parabola at R.

How do you show that the tangent at P is x₁x = 2a(y+y₁)

PLEASE IF ANYONE COULD HELP ME SOLVE THIS QUESTION AND EXPLAIN HOW THIS RELATES TO THE CHORD OF CONTACT
I WILL BE SO GRATEFUL

THANK YOU <3

 

[andsyd96]

the final solution doesnt look right...

 

[Carrotsticks]

Hi Guys,

[An alternative proof of the reflection property] P is the variable point (x₁, y₁) on the parabola x² = 4ay, and S is the focus. The tangent at P meets the tangent at the vertex of the parabola at Q an it meets the axis of the parabola at R.

How do you show that the tangent at P is x₁x = 2a(y+y₁)

PLEASE IF ANYONE COULD HELP ME SOLVE THIS QUESTION AND EXPLAIN HOW THIS RELATES TO THE CHORD OF CONTACT
I WILL BE SO GRATEFUL

THANK YOU <3

Make y the subject, so y=x^2/4a.

Differentiate with respect to x to get y'=x/2a.

Substitute x=x_1 to get the gradient m=x_1/2a.

Now use the point-gradient formula with the gradient m and the point P(x_1,y_1)

 

[Bobbo1]

Make y the subject, so y=x^2/4a.

Differentiate with respect to x to get y'=x/2a.

Substitute x=x_1 to get the gradient m=x_1/2a.

Now use the point-gradient formula with the gradient m and the point P(x_1,y_1)

+1

 

[yeyjasminetime]

Make y the subject, so y=x^2/4a.

Differentiate with respect to x to get y'=x/2a.

Substitute x=x_1 to get the gradient m=x_1/2a.

Now use the point-gradient formula with the gradient m and the point P(x_1,y_1)

Thank you again x

 

[yeyjasminetime]

Make y the subject, so y=x^2/4a.

Differentiate with respect to x to get y'=x/2a.

Substitute x=x_1 to get the gradient m=x_1/2a.

Now use the point-gradient formula with the gradient m and the point P(x_1,y_1)

So I did y - y_1 = x_1/2a ( x- x_1)

then I get 2ay-2ay_1 = x_1x - x_1^2 by expanding

then x_1x = 2ay-2ay_1 + x_1^2

but I still cant get x1x = 2a(y+y1)

 

[yeyjasminetime]

Acutally nevermind i get it!!

 

[yeyjasminetime]

Does anyone know how to prove that a normal does not pass through the focus?

(if you would like the question just let me know)

x

 

[HeroicPandas]

Find normal
Sub focus in
Show that LHS=/=RHS
This means focus does not satisfy equation, hence it NORMAL does not pass through focus

I cannot understand how u cannot do this

 

[yeyjasminetime]

Find normal
Sub focus in
Show that LHS=/=RHS
This means focus does not satisfy equation, hence it NORMAL does not pass through focus

I cannot understand how u cannot do this

What do you mean by RHS and LHS?

 

[yeyjasminetime]

Find normal
Sub focus in
Show that LHS=/=RHS
This means focus does not satisfy equation, hence it NORMAL does not pass through focus

I cannot understand how u cannot do this

So x + py = 2ap + ap^3

Sub in (0,a)

0 +pa = 2ap +ap^3

and then??

 

[HeroicPandas]

Say ze normale is y = x

The focus is (100, 1)

Sub in

1 = 100

Does LHS (left-hand side) = RHS (right-hand side)?

Since it doesnt, normale doesnt hit the focus

 

[asianese]

The normal is

 

[yeyjasminetime]

the normal is

thank you omg