Question

Find the coordinates of the points of intersection of the line y=3x-1 and the parabola $y^2=4x$

Answer 1

The abscissae of the points of intersection of the line $y=3x-1$ and the parabola $y^2=4x$ are ratio of equation.

${(3x-1)}^2=4x$

$\Rightarrow {9x}^2+1-6x=4x$

$\Rightarrow {9x}^2-6x-4x+1=0$

$\Rightarrow {9x}^2-10x+1=0$

$\Rightarrow {9x}^2-9x-x+1=0$

$\Rightarrow 9x(x-1)-1(x-1)=0$

$x=1$ $x=\frac{1}{9}$

Substituting $x=1$ and $\frac{1}{9}$ in $y=3x-1$ we get $y=2$ and $y=\frac{-2}{3}$ points of intersection $(1,2)$ and $(\frac{1}{9},\frac{-2}{3})$.