Find the equation PQ, then find pq, by using the equation of the line and the point (0,a), and then use the distance formula.
I cant seem to get this one out:
P(2ap,ap^{2}) & Q(2aq,aq^{2}) lie on the parabola x^{2} = 4ay, where a > 0. The chord PQ passes through the focus.
Show that the chord PQ has length A(p + 1/p)^{2}
thanks for your help
Find the equation PQ, then find pq, by using the equation of the line and the point (0,a), and then use the distance formula.
Is quite inactive.
yeh i can find everything except it doesnt seem to work out when i do it with the distance formula.
i've already shown that pq = -1
but my working gets really messy with the distance formula and just doesnt solve :<
yup lol, i remembered that.
can somebody give the algebra a go?
Length = sqrt[(x2 - x1)^2 + (y2 - y1)^2]Originally Posted by elseany
=sqrt[(2ap -2aq)^2 + (ap^2 - aq^2)^2]
=sqrt[4a^2(p - q)^2 + a^2(p^2 - q^2)^2]
=sqrt[4a^2(p + 1/p)^2 + a^2(p^2 - 1/p^2)^2]
=a sqrt[4(p + 1/p)^2 + (p^2 - 1/p^2)^2]
=a sqrt[(p + 1/p)^2 (4 + (p - 1/p)^2)]
=a sqrt[(p + 1/p)^2 (p + 1/p)^2]
=a(p + 1/p)^2
Or:
let S be the focus (0,a), we know that SP=ap^2 + a = a(p^2 + 1)
SQ = aq^2 + a = a(1/p^2 + 1) [think why?]
so PQ = QS + SP = a (p^2 + 2 + 1/p^2) = a(p + 1/p)^2
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thanks jyu and affinity, but theres something i dont get;
=a sqrt[(p + 1/p)^2 (4 + (p - 1/p)^2)]
=a sqrt[(p + 1/p)^2 (p + 1/p)^2]
how do you go from that first line to that second line?
Try expanding (4 + (p - 1/p)^2)] and it should turn out to be the same as (p + 1/p)^2]
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