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[Da_Bomb]

1. show that the locus of the mid point of chords in the parabola xsquared=4ay and which pass through the vertex is another parabola xsquared=2ay.

2. Two points, P , Q move on the parabola xsquared=4ay so that the x coordinates of P and Q differ by a constant value, 2a. What is the locu of M. the mid point of PQ.

3. Prove that the locus of the mid point M of focal chords in the parabola xsquared=4ay is the parabola xsquared=2a(y-a).

 

[kaz1]

1. Let P(2at,at²) be a point on the parabola
P makes a chord on the parabola with the origin (which is the vertex) O(0,0).
Midpoint of PO is M(at,at²/2).
We find the locus of M by eliminating t.
x=at
t=x/a
.'. y= a.x²/a².1/2
x²=2ay

 

[kaz1]

For 2 let the x coordinate P be 2at and for Q 2at+2a or 2at-2a.

 

[gurmies]

Let P and Q be points on the parabola, with coordinates (2ap, ap^2) and (2aq, aq^2) respectively.

Equation of chord is:

y - ap^2 = (aq^2 - ap^2)(x - 2ap)/(2aq - 2ap)

y - ap^2 = a(q - p)(q + p)(x - 2ap)/2a(q - p)

y - ap^2 = (q + p)(x - 2ap)/2

Since this is a focal chord, (a, 0) satisfies the equation.

-2ap^2 = qa - 2apq + pa - 2ap^2

[2pq = p + q]

Now, midpoint of chord PQ is given by:

x = (2ap + 2aq)/2 ===> a(p+q)

y = (ap^2 + aq^2)/2 ===> a(p^2 + q^2)/2

p + q = x/a

y = a/2 x [(p + q)^2 - 2pq]

= a/2 x [(x/a)^2 - x/a]

= a/2 x [(x^2/a^2) - x/a]

= a/2(x^2 - ax)/a^2

2ay = x^2 - ax

...and I've made an error somewhere.

 

[jet]

Let P and Q be points on the parabola, with coordinates (2ap, ap^2) and (2aq, aq^2) respectively.

Equation of chord is:

y - ap^2 = (aq^2 - ap^2)(x - 2ap)/(2aq - 2ap)

y - ap^2 = a(q - p)(q + p)(x - 2ap)/2a(q - p)

y - ap^2 = (q + p)(x - 2ap)/2

Since this is a focal chord, (a, 0) satisfies the equation.

-2ap^2 = qa - 2apq + pa - 2ap^2

[2pq = p + q]

Now, midpoint of chord PQ is given by:

x = (2ap + 2aq)/2 ===> a(p+q)

y = (ap^2 + aq^2)/2 ===> a(p^2 + q^2)/2

p + q = x/a

y = a/2 x [(p + q)^2 - 2pq]

= a/2 x [(x/a)^2 - x/a]

= a/2 x [(x^2/a^2) - x/a]

= a/2(x^2 - ax)/a^2

2ay = x^2 - ax

...and I've made an error somewhere.

Focal chord goes through (0,a). Then you get pq = -1 and it follows.

 

[gurmies]

Focal chord goes through (0,a). Then you get pq = -1 and it follows.

Where the hell did I get (a,0) from?...this is what happens when you abstain from maths for even one week.

 

[jet]

I know, I lose it too.