Answer:
[tex]1. P(3,0); 2. P(-1,-\frac{12}{5} ); 3. P(\frac{16}{7},\frac{27}{7} ); 4.P(-3,-6)[/tex]
Step-by-step explanation:
If coordinates of points are [tex]A(x_1,y_1)[/tex] and [tex]B(x_2,y_2)[/tex] and you need to divide the segment AB in the ratio m:n, where [tex]\lambda=\frac{m}{n}[/tex], then coordinates of point P which partitions segment AB in given ratio are:
[tex]x=\frac{x_1+ \lambda x_2}{1+\lambda}, y=\frac{y_1+\lambda y_2}{1+\lambda}[/tex]
1. Here, [tex]A(-3,-2), B(6,1), m:n=2:1,[/tex] which means [tex]\lambda=\frac{2}{1}=2[/tex], so we get the following coordinates:
[tex]x=\frac{-3+2*6}{1+2}=3 , y=\frac{-2+2*1}{1+2}=0[/tex]
We got that point P has coordinates P(3,0).
2. Here, [tex]A(-3,-4), B(2,0), m:n=2:3,[/tex] which means [tex]\lambda=\frac{2}{3},\\[/tex] so, we get the following coordinates:
[tex]x=\frac{-3+\frac{2}{3}*2 }{1+\frac{2}{3} }=\frac{\frac{-9+4}{3} }{\frac{5}{3} }=-1,\\y=\frac{-4+\frac{2}{3}*0 }{1+\frac{2}{3} }=\frac{-\frac{4}{1} }{\frac{5}{3} }=-\frac{12}{5}[/tex]
We got that point P has coordinates [tex]P(-1,-\frac{12}{5} )[/tex].
3. Here, [tex]A(-2,1), B(-4,5), m:n=5:2,[/tex] which means [tex]\lambda=\frac{5}{2}[/tex], so we get the following coordinates:
[tex]x=\frac{-2+\frac{5}{2}*4 }{1+\frac{5}{2} }=\frac{\frac{-4+20}{2} }{\frac{7}{2} }=\frac{16}{7},\\y=\frac{1+\frac{5}{2}*5 }{1+\frac{5}{2} }=\frac{\frac{2+25}{2} }{\frac{7}{2} }=\frac{27}{7}[/tex]
We got that point P has coordinates [tex]P(\frac{16}{7} ,\frac{27}{7} )[/tex].
4. Here, [tex]A(-9,-9), B(5,-2), m:n=3:4,[/tex] which means [tex]\lambda=\frac{3}{4}[/tex] so we get the following coordinates:
[tex]x=\frac{-9+\frac{3}{4}*5 }{1+\frac{3}{4} }=\frac{-\frac{36+15}{4} }{\frac{7}{4} }=-3,\\y=\frac{-9+\frac{3}{4}(-2) }{1+\frac{3}{4} }=\frac{-\frac{36-6}{4} }{\frac{7}{4} }=-6\\[/tex]
We got that point P has coordinates P(-3,-6).
