The points of intersection of l x + m y = 1 lx+my=1 l x + m y = 1 and y 2 = 4 a x y^2=4ax y 2 = 4 a x are:
l x + m y = 1 ⇒ m y = 1 − l x ⇒ y = 1 − l x m lx+my=1\Rightarrow my=1-lx\Rightarrow y=\dfrac{1-lx}m l x + m y = 1 ⇒ m y = 1 − l x ⇒ y = m 1 − l x
( 1 − l x m ) 2 = 4 a x ⇒ 1 − 2 l x + l 2 x 2 m 2 = 4 a x ⇒ \left( \dfrac{1-lx}m\right)^2=4ax\Rightarrow \dfrac{1-2lx+l^2x^2}{m^2}=4ax\Rightarrow ( m 1 − l x ) 2 = 4 a x ⇒ m 2 1 − 2 l x + l 2 x 2 = 4 a x ⇒
l 2 x 2 + ( − 2 l − 4 a m 2 ) x + 1 = 0 ⇒ x = l + 2 a m 2 ± 2 m a ( l + a m 2 ) l 2 l^2x^2+(-2l-4am^2)x+1=0\Rightarrow x=\dfrac{l+2am^2\pm 2m\sqrt{a(l+am^2)}}{l^2} l 2 x 2 + ( − 2 l − 4 a m 2 ) x + 1 = 0 ⇒ x = l 2 l + 2 a m 2 ± 2 m a ( l + a m 2 )
⇒ y = − 2 a m ∓ 2 a ( l + a m 2 ) \Rightarrow y=-2am\mp 2\sqrt{a(l+am^2)} ⇒ y = − 2 am ∓ 2 a ( l + a m 2 )
The points of intersection are:
P ( l + 2 a m 2 + 2 m a ( l + a m 2 ) l 2 , − 2 a m − 2 a ( l + a m 2 ) ) = O P → P\left( \dfrac{l+2am^2+2m\sqrt{a(l+am^2)}}{l^2}, -2am-2\sqrt{a(l+am^2)}\right)=\overrightarrow{OP} P ( l 2 l + 2 a m 2 + 2 m a ( l + a m 2 ) , − 2 am − 2 a ( l + a m 2 ) ) = OP
Q ( l + 2 a m 2 − 2 m a ( l + a m 2 ) l 2 , − 2 a m + 2 a ( l + a m 2 ) ) = O Q → Q\left( \dfrac{l+2am^2-2m\sqrt{a(l+am^2)}}{l^2}, -2am+2\sqrt{a(l+am^2)}\right)=\overrightarrow{OQ} Q ( l 2 l + 2 a m 2 − 2 m a ( l + a m 2 ) , − 2 am + 2 a ( l + a m 2 ) ) = OQ
We know that the lines joining the origin O O O are at right angles: O P → ⋅ O Q → = 0 : \overrightarrow{OP}\cdot \overrightarrow{OQ}=0: OP ⋅ OQ = 0 :
( l + 2 a m 2 + 2 m a ( l + a m 2 ) ) ( l + 2 a m 2 − 2 m a ( l + a m 2 ) ) l 2 ⋅ l 2 + \dfrac{(l+2am^2+2m\sqrt{a(l+am^2)})(l+2am^2-2m\sqrt{a(l+am^2)})}{l^2\cdot l^2}+ l 2 ⋅ l 2 ( l + 2 a m 2 + 2 m a ( l + a m 2 ) ) ( l + 2 a m 2 − 2 m a ( l + a m 2 ) ) +
+ ( − 2 a m − 2 a ( l + a m 2 ) ) ( − 2 a m + 2 a ( l + a m 2 ) ) = +(-2am-2\sqrt{a(l+am^2)})(-2am+2\sqrt{a(l+am^2)})= + ( − 2 am − 2 a ( l + a m 2 ) ) ( − 2 am + 2 a ( l + a m 2 ) ) =
= 1 l 2 − 4 l a = 0 ⇒ 1 l 2 = 4 l a ⇒ 1 = 4 l a l 2 ⇒ 1 = 4 l 3 a =\dfrac{1}{l^2}-4la=0 \Rightarrow \dfrac{1}{l^2}=4la \Rightarrow 1=4lal^2\Rightarrow \boxed{1=4l^3a} = l 2 1 − 4 l a = 0 ⇒ l 2 1 = 4 l a ⇒ 1 = 4 l a l 2 ⇒ 1 = 4 l 3 a
We see that the problem statement is incorrect. It is correct that 4 a l 3 = 1 4al^3=1 4 a l 3 = 1 is satisfied.
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