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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 :smile:
 

Carrotsticks

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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 :smile:
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).
 
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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
 

Carrotsticks

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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

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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
 
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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
 
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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)
 
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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

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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
 
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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

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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
 

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