Answer:
The coordinates of point B is [tex](\frac{21}{4} , 2 )[/tex]
Step-by-step explanation:
Given:
Let,
[tex]B \equiv (x,y)\\A \equiv (x1,y1) \equiv (7,4)\\C \equiv (x2,y2) \equiv (0,-4)[/tex]
[tex]\frac{AB}{AC} =\frac{1}{4}[/tex]
First we need to find [tex]\frac{AB}{BC}[/tex]
[tex]\therefore \frac{AB}{AC} = \frac{1}{4}\\\therefore \frac{AC}{AB} = \frac{4}{1}\ Invertendo\\\therefore \frac{AC-AB}{AB} = \frac{4-1}{1}\ Dividendo\\ \therefore \frac{BC}{AB} = \frac{3}{1}\\ \therefore \frac{AB}{BC} = \frac{1}{3}\ Invertendo\\\therefore \frac{AB}{BC} = \frac{1}{3} = \frac{m}{n}\ say[/tex]
Now point B divide segment AC internally in the ratio m : n i.e 1/3.
So, by internal division formula, the X coordinate and the Y coordinate of point B are as follow
[tex]x =\frac{mx2+nx1}{m+n}\ and\ y = \frac{my2+ny1}{m+n}\\x =\frac{1\times 0 + 3\times 7}{1+3}\ and\ y =\frac{1\times -4 + 3\times 4}{1+3}\\x =\frac{21}{4}\ and\ y =\frac{8}{4}\\x =\frac{21}{4}\ and\ y = 2[/tex]
Therefore,The coordinates of point B is [tex](\frac{21}{4} , 2 )[/tex]
