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ilovealgebra

Member

Registered: 2006-10-02

Posts: 40

calculus

A parabola is given by y^2=4ax and the ellipse (x^2)/a^2  +  (y^2)/b^2  =  1 where a>0 and b>0, meet at the points P and Q.

(a) The two curves intersect in such a way that the tangent to the parabola at P is perpendicular to the tangent tothe ellipse at P.

(i) Show that B^2 = 2a^2
(ii) Hence, find in terms of a, the distance of the point P from the origin O.

(b) The tangent to the parabola at P meets the x-axis at M. The tangent to the ellipse at P meets the x-axis at N. Show that the length of MN=(2*(sqrt)(2))a.

Hi guys, I have solved (a), but am having problems with (b) I was wondering if anyone can show a full solution using a parametric method. i.e. using (at^2,2at) for the parabola and dy/dx=(1/t) if that helps. Cheers

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"...nothing physical which sense-experience sets before our eyes, or which necessary demonstrations prove to us, ought to be called into question (much less condemned) upon the testimony of biblical passages."
-Galileo Galilei

JaneFairfax

Member

Registered: 2007-02-23

Posts: 6,868

Re: calculus

Assuming that P is in the first quadrant, I make it that it’s the point

.

The tangent to the parabola at P is

and the tangent to the ellipse at P is

. It is a matter of simple algebra to show that given straight lines of nonzero gradients

and

intersecting at

, the distance between their

-intercepts is

.

Last edited by JaneFairfax (2009-09-13 17:59:14)

jennyLOVESmaths

Member

Registered: 2009-07-13

Posts: 1

Re: calculus

for algebra,
what does x replace?

JaneFairfax

Member

Registered: 2007-02-23

Posts: 6,868

Re: calculus

Sorry, I don’t understand you.

ilovealgebra

Member

Registered: 2006-10-02

Posts: 40

Re: calculus

[math]\pi(n) = \sum_{m=2}ˆ{n}
math]

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"...nothing physical which sense-experience sets before our eyes, or which necessary demonstrations prove to us, ought to be called into question (much less condemned) upon the testimony of biblical passages."
-Galileo Galilei

ilovealgebra

Member

Registered: 2006-10-02

Posts: 40

Re: calculus

Woops, was trying to figure latex and accidently pressed enter, sorry

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"...nothing physical which sense-experience sets before our eyes, or which necessary demonstrations prove to us, ought to be called into question (much less condemned) upon the testimony of biblical passages."
-Galileo Galilei

bobbym

bumpkin

From: Bumpkinland

Registered: 2009-04-12

Posts: 109,606

Re: calculus

Hi ilovealgebra;

I think you will need some more to describe the prime counting function. It is the number of primes less than or equal to n.

π(10) = 4 , {2,3 5, 7}

π(100) = 25

π(10000) = 1229

π(10 000 000 000) = 455 052 511

Famous in number theory. Go here:

http://en.wikipedia.org/wiki/Prime-counting_function

Last edited by bobbym (2009-09-15 07:58:49)

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In mathematics, you don't understand things. You just get used to them.
If it ain't broke, fix it until it is.
Always satisfy the Prime Directive of getting the right answer above all else.