Question

Show that the line x + y = 4 intersects the parabola $y^2=2x$ at two points P and Q. Find the coordinates of these points and the length PQ.

Answer 1

given $x+y=4$ intersects the parabola at P and Q

$\Rightarrow$ the points must satisfy both equations

therefore

$y=4-x$ substituting in $y^2=2x$ we get those coordinates

$\Rightarrow (x-4{)}^2=2x$

$x^2-10x+16=0$ the solutions are $x=\frac{10+\sqrt{100-64}}{2},\frac{10-\sqrt{100-64}}{2}$

$i.e$ x=8,2

substituting in $x+y=4$equation we get $y=-4,2$ respectively

therefore the points are $P(8,-4),Q(2,2)$