A line with direction ratio 2,1,2 meets the lines x=y+2=z and x+2=2y=2z respectively at the point P and Q. if the length of the perpendicular from…

Let $P(t,t-2,t)$P(t,t−2,t) and $Q(2s-2,s,s)$Q(2s−2,s,s)

$\mathrm{DR}'s$DR′s of $\mathrm{PQ}$PQ are $(2,1,2)$(2,1,2)

$$\Rightarrow \frac{2s-2-t}{2}=\frac{s-t+2}{1}=\frac{s-t}{2}$$⇒22s−2−t​=1s−t+2​=2s−t​

$$\Rightarrow \frac{2s-2-t}{2}=\frac{s-t+2}{1}$$⇒22s−2−t​=1s−t+2​

$$\Rightarrow 2s-2-t=2s-2t+4$$⇒2s−2−t=2s−2t+4

$$\Rightarrow t=6$$⇒t=6

$$\Rightarrow \frac{s-t+2}{1}=\frac{s-t}{2}$$⇒1s−t+2​=2s−t​

$$\Rightarrow \frac{s-6+2}{1}=\frac{s-6}{2}$$⇒1s−6+2​=2s−6​

$$\Rightarrow 2s-8=s-6$$⇒2s−8=s−6

$$\Rightarrow s=2$$⇒s=2

$\Rightarrow P(6,4,6)$⇒P(6,4,6) and $Q(2,2,2)$Q(2,2,2)

$$\mathrm{PQ}:\frac{x-2}{2}=\frac{y-2}{1}=\frac{z-2}{2}=\lambda$$PQ:2x−2​=1y−2​=2z−2​=λ

$$\Rightarrow x=2\lambda +2,y=\lambda +2\text{and}z=2\lambda +2$$⇒x=2λ+2,y=λ+2andz=2λ+2

Let, $F(2\lambda +2,\lambda +2,2\lambda +2)$F(2λ+2,λ+2,2λ+2)

$$A(1,2,12)$$A(1,2,12)

$$\vec{\mathrm{AF}}\cdot \vec{\mathrm{PQ}}=0$$AF⋅PQ​=0

$$\Rightarrow 2(2\lambda +1)+1(\lambda )+2(2\lambda -10)=0$$⇒2(2λ+1)+1(λ)+2(2λ−10)=0

$$\Rightarrow 4\lambda +2+\lambda +4\lambda -20=0$$⇒4λ+2+λ+4λ−20=0

$$\Rightarrow \lambda =2$$⇒λ=2

So $F(6,4,6)$F(6,4,6) and $\mathrm{AF}=\sqrt{65}$AF=65​

$$\Rightarrow {l}^{2}=65$$⇒l2=65