Question #223986

The lines joining the origin to the points of intersection of the line lx + my = 1 and y2= 4ax are at right angles. Show that 4al = 1.


1
Expert's answer
2022-01-31T13:15:35-0500

The points of intersection of lx+my=1lx+my=1 and y2=4axy^2=4ax are:

lx+my=1my=1lxy=1lxmlx+my=1\Rightarrow my=1-lx\Rightarrow y=\dfrac{1-lx}m

(1lxm)2=4ax12lx+l2x2m2=4ax\left( \dfrac{1-lx}m\right)^2=4ax\Rightarrow \dfrac{1-2lx+l^2x^2}{m^2}=4ax\Rightarrow

l2x2+(2l4am2)x+1=0x=l+2am2±2ma(l+am2)l2l^2x^2+(-2l-4am^2)x+1=0\Rightarrow x=\dfrac{l+2am^2\pm 2m\sqrt{a(l+am^2)}}{l^2}

y=2am2a(l+am2)\Rightarrow y=-2am\mp 2\sqrt{a(l+am^2)}

The points of intersection are:

P(l+2am2+2ma(l+am2)l2,2am2a(l+am2))=OPP\left( \dfrac{l+2am^2+2m\sqrt{a(l+am^2)}}{l^2}, -2am-2\sqrt{a(l+am^2)}\right)=\overrightarrow{OP}

Q(l+2am22ma(l+am2)l2,2am+2a(l+am2))=OQQ\left( \dfrac{l+2am^2-2m\sqrt{a(l+am^2)}}{l^2}, -2am+2\sqrt{a(l+am^2)}\right)=\overrightarrow{OQ}

We know that the lines joining the origin OO are at right angles: OPOQ=0:\overrightarrow{OP}\cdot \overrightarrow{OQ}=0:

(l+2am2+2ma(l+am2))(l+2am22ma(l+am2))l2l2+\dfrac{(l+2am^2+2m\sqrt{a(l+am^2)})(l+2am^2-2m\sqrt{a(l+am^2)})}{l^2\cdot l^2}+

+(2am2a(l+am2))(2am+2a(l+am2))=+(-2am-2\sqrt{a(l+am^2)})(-2am+2\sqrt{a(l+am^2)})=

=1l24la=01l2=4la1=4lal21=4l3a=\dfrac{1}{l^2}-4la=0 \Rightarrow \dfrac{1}{l^2}=4la \Rightarrow 1=4lal^2\Rightarrow \boxed{1=4l^3a}

We see that the problem statement is incorrect. It is correct that 4al3=14al^3=1 is satisfied.


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