1118195562 2004 Mathematics Extension 2 Notes Nathan Ernest & J.B Fitzpatrick

C

l 

B

h   A

 x 

D

If a circle is constructed such that the line drawn back from the try point is a tangent to the circle, then the point where the circle and the line touch is the position to give the greatest angle angle at the goal mouth. mouth. Any Any point point outsid outsidee the circle circle will give give an angle smaller smaller,, and any point inside the circle will give an angle larger, but there is no point that lies inside the circle and lies along this line. Therefore, using the intercept theorem, we say 2

x

=  h × (h + l)

x  =

∴

 

h2

+ hl

However, using calculus is a bit more tricky. h

Let ∠ADB =  α and ∠BD C  =  θ , therefore tan α =  and tan(α + θ) = x Now we follow follow these mathematical mathematical processes. processes. tan α + tan θ tan(α + θ) = 1 − tan α tan θ h  + tan θ l + h x ∴ = θ x 1 − h tan x Algebra manipulation leads us to, tan θ  =

lx hl + h2

+ x2

We differentiate both sides of this equation with respect to x. 2

sec let

dθ dx

=0

θ

∴

dθ ·

dx

=



hl + h2

+ x2



2

=  hl  + h2

∴  x  =

−

lx (2x)

(lh + h2 + x2 )2

2lx2 =  hl 2 + h2 l + lx2 ∴  x

l

 

hl + h2

1

l  + h x

.