a) Chcemy

$\left|AB\right|=10$

Zatem 

$\sqrt{{\left(p-\left(-4\right)\right)}^2+{\left(-2-6\right)}^2}=10$  

${\left(p+4\right)}^2+{\left(-8\right)}^2={10}^2$  

${\left(p+4\right)}^2+64=100$  

${\left(p+4\right)}^2=36$  

$p+4=6\text{lub}p+4=-6$  

$p=2\text{lub}p=-10$  

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b) Chcemy

$\left|AB\right|=8$

Zatem 

$\sqrt{{\left(-1-5\right)}^2+{\left(-3-p\right)}^2}=8$  

${\left(-6\right)}^2+{\left(-3-p\right)}^2=8^2$  

$36+{\left(-3-p\right)}^2=64$

${\left(-3-p\right)}^2=28$

$-3-p=2\sqrt{7}\text{lub}-3-p=-2\sqrt{7}$  

$-p=2\sqrt{7}+3\text{lub}-p=-2\sqrt{7}+3$  

$p=-2\sqrt{7}-3\text{lub}p=2\sqrt{7}-3$  

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c) Chcemy

$\left|AB\right|=6$

Zatem 

$\sqrt{{\left(p-2\right)}^2+{\left(-p-4\right)}^2}=6$  

${\left(p-2\right)}^2+{\left(p+4\right)}^2=6^2$  

$p^2-4p+4+p^2+8p+16=36$

$2p^2+4p+20=36$  

$2p^2+4p-16=0$  

$p^2+2p-8=0$  

${\Delta }_p=2^2-4\cdot 1\cdot \left(-8\right)=4+32=36,sqr\left\{{\Delta }_p\right\}=6$  

$p=\frac{-2-6}{2}\text{lub}p=\frac{-2+6}{2}$  

$p=-4\text{lub}p=2$  

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d) Chcemy

$\left|AB\right|=2$

Zatem 

$\sqrt{{\left(3-p\right)}^2+{\left(p+1-2p\right)}^2}=2$  

${\left(3-p\right)}^2+{\left(1-p\right)}^2=2^2$  

$9-6p+p^2+1-2p+p^2=4$

$2p^2-8p+10=4$  

$2p^2-8p+6=0$  

${\Delta }_p={\left(-8\right)}^2-4\cdot 2\cdot 6=64-48=16,\sqrt{{\Delta }_p}=4$  

$p=\frac{8-4}{4}\text{lub}p=\frac{8+4}{4}$  

$p=1\text{lub}p=2$