Find the equation of the normal to the curve $y= \frac{x-2}{1+2x}$ (1) at the point where the curve cuts the $x$-axis . Find the coordinates of the point where this normal cuts the curve again .
I found The equation of the normal –
$ y = -5x+10$ (2)
Now , I use simultaneous equation to find the coordinates .
I sub equation 2 to 1 .
I eventually get a quadratic equation –
$-10x^2 + 14x + 12 = 0 $
$x = 2 , \frac{-3}{5} $
I'm shocked now because I'm not sure which one to reject and why ?
Or do I not reject it and sub both of this x values to find 2 values of y meaning I have 2 coordinates ? Or do I have to reject ? Thanks !
Best Answer
Since the normal passes through the point $(2,0)$ this is also an intersection point between the line and the curve, corresponding to your solution $x=2$. The other value of $x$ that you have found gives the other intersection.