Question

If the tangent to the parabola $y=x(2-x)$ at the point (1, 1) intersects the parabola at P. Find the coordinate of P.

Answer 1

The equation of parabola is
$y=x(2-x)$....(i)
On differentiating, w.r.t. x, we get
$\frac{dy}{dx}=2-2x$
Slope of tangent at (1, 1).
$m=2-2\left(1\right)=0$
$\therefore$ Equation of tangent at (1, 1) is
$(y-1)=0(x-1)$
$\Rightarrow y-1=0$....(ii)
On solving Eqs. (i) and (ii), we get
$\Rightarrow 2x-x^2=1$
$\Rightarrow x^2-2x+1=0$
$\Rightarrow x=1$
$\therefore$
coordinates of point $P$ are $(1,1)$.