Question

A line of slope 2 passes through the point A(1,3).
a) Check whether B(3,7) is a point on this line.
b) Write down the equation of this line.
c) Find the coordinates of a point C on the line such that BC = 2AB.

Answer 1

Slope of the line is 2 and passes through A(1, 3)
$\frac{y-3}{x-1}=2$
$y-3=2x-2$
$y=2x+1$
(a) For point B(3, 7)
2(3) + 1 = 7
Hence the point B(3, 7)lies on the line y = 2x + 1
(b) Equation of the line is y = 2x+1.
(c) Let $(x_1,y_1)$ be the coordinates of point C on line.
Therefore it satisfies the equation of line a $y=2x+1$
$\therefore y_1=2x_1+1$
$BC=2AB$
$\Rightarrow \sqrt{(x_1-3{)}^2+(y_1-7{)}^2}=2\sqrt{(3-1{)}^2+(7-3{)}^2}$
Squaring both sides, we get
$\Rightarrow (x_1-3{)}^2+(y_1-7{)}^2=4\left[\right(3-1{)}^2+(7-3{)}^2]$
$(x_1-3{)}^2+(2x_1+1-7{)}^2=4\left[\right(2{)}^2+\left(4{)}^2\right]$ $[\therefore y_1=2x_1+1]$
$5(x_1-3{)}^2=80$
$(x_1-3{)}^2=16$
$x_1-3=\pm 4$
$x_1=7$ or $-1$
$y_1=2x_1+1=2\left(7\right)+1=15$
$y_1=2x_1+1=2(-1)+1=-1$
The coordinates of point C are $(7,15)$ or $(-1,-1)$.