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another tangent problem (1 Viewer)

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tangents from the pt P(m,n) touch the parabola x^2=8y at the pts A and B. show that the x-coordinates of A and B are the roots of the quadratic equation x^2-2mx+8n=0
 

Riviet

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Substitute P into chord of contact: xx0 = 2a(y + yo) and note that a=2 since x2=4ay
mx=4(y+n)
y=mx/4 - n
But y=x2/8

.'. mx/4 - n=x2/8
Then multiply by 8 and bring everything to RHS to get required result.
 
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SoulSearcher

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Find the equation of the chord of contact:
xx0 = 2a(y+y0, a = 2, P(m,n)
mx = 4(y+n)
y = (mx-4n)/4

Sub into the equation x2=8y to find x-coordinates
x2 = 8[(mx-4n)/4)
x2 = 2mx - 8n
x2 - 2mx + 8n = 0

Therefore x-coordinates of A and B are the roots of the quadratic equation x2[/sup-2mx+8n = 0
 

Mumma

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You dont need equation for chord of contact
Simply,

dy/dx = x/4

tangent at point (t,t^2/8)
y - t^2/8 = t/4(x-t)
8y - t^2 = 2t(x-t) goes through (m,n)
8n - t^2 = 2tm - 2t^2
t^2 - 2tm + 8n = 0

Since t is the x coordinate,
x^2 - 2mx + 8n = 0 has roots that are the x-coordinates of A,B
 

airie

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Eww. Calculus. <.<

I think I like "elementary methods" better XP
 

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