another tangent problem (1 Viewer)

[lollol]

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]

Substitute P into chord of contact: xx₀ = 2a(y + y_o) 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.

 

[SoulSearcher]

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 x^{2[/sup-2mx+8n = 0}

 

[Mumma]

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]

Eww. Calculus. <.<

I think I like "elementary methods" better XP

 

[SoulSearcher]

Eww. Calculus. <.<

I think I like "elementary methods" better XP

I concur