$$ (Q-5,2) A B $$ kesmada $$ A(-8 ; 6) $$ va $$ B(-2 ;-3) $$. Shu $$ A B $$ kesmada E nuqta shunday Olnarki. $$ 2 A E=3 E B $$. $$ E $$ nuqta koordinatalarini toping.

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1. Berilgan: $$ A(-8,\,6) $$ va $$ B(-2,\,-3) $$. E nuqtasi togʻridan-toʻgʻri $$ AB $$ kesmada joylashgan va shartga koʻra  
   $$
   2\cdot AE = 3\cdot EB.
   $$

2. Shartni $$ AE $$ va $$ EB $$ nisbati sifatida ifodalash uchun, har ikki tomonini $$ EB $$ ga bo‘lamiz:  
   $$
   \frac{AE}{EB} = \frac{3}{2}.
   $$
   Demak, E nuqta $$ AB $$ kesmani $$ 3:2 $$ nisbatida bo‘lib beradi, yaʼni  
   $$
   AE:EB = 3:2.
   $$

3. Ichki bo‘linish formulasiga ko‘ra, agar nuqta $$ E $$ kesmani $$ m:n $$ nisbatida bo‘lishi shart bo‘lsa (yaʼni $$ AE:EB = m:n $$), uning koordinatalari quyidagicha ifodalanadi:  
   $$
   E\left(\frac{n\cdot A_x + m\cdot B_x}{m+n},\, \frac{n\cdot A_y + m\cdot B_y}{m+n}\right).
   $$
   Bizda $$ m=3 $$ va $$ n=2 $$.

4. Shu formuladan foydalanib, $$ E $$ nuqtaning $$ x $$ va $$ y $$ koordinatalarini hisoblaysiz:
   - $$ x $$-koordinata:
     $$
     E_x=\frac{2\cdot (-8) + 3\cdot (-2)}{3+2}=\frac{-16 - 6}{5}=\frac{-22}{5}.
     $$
   - $$ y $$-koordinata:
     $$
     E_y=\frac{2\cdot 6 + 3\cdot (-3)}{3+2}=\frac{12 - 9}{5}=\frac{3}{5}.
     $$

5. Natija:  
   $$
   E\left(-\frac{22}{5},\,\frac{3}{5}\right).
   $$
E nuqta koordinatalari $$ \left(-\frac{22}{5},\,\frac{3}{5}\right) $$.

kesmada va . Shu kesmada E nuqta shunday
Olnarki. . nuqta koordinatalarini toping.

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kesmada va . Shu kesmada E nuqta shunday Olnarki. . nuqta koordinatalarini toping.