1. ## parabola problems

1- prove that the line X+Y=1 touches the parabola Y=X-X²
and what is the coordinates of tangency

2-find the equation of the parabola having 3X+4Y-5=0 as directrix and (5,-6) as focus find it is vertex latux rectum and the equation of its axis and its tangent at the vertex

2. Hello, mohamedsafy!

1) Prove that the line $\displaystyle x + y \:=\:1$ touches the parabola $\displaystyle y\:=\:x-x^2$

and find the coordinates of tangency.
A line can intersect a parabola in 0, 1 or 2 points.

We have: .$\displaystyle \begin{array}{cccc}y &=& 1-x & {\color{blue}[1]}\\ y &=&x -x^2 & {\color{blue}[2]}\end{array}$

Equate [1] and [2]: .$\displaystyle 1 - x \:=\:x-x^2\quad\Rightarrow\quad x^2-2x + 1 \:=\:0 \quad\Rightarrow\quad (x-1)^2 \:=\:0$

. . $\displaystyle x-1\:=\:0 \quad\Rightarrow\quad x \:=\:1$

Substitute into [1]: .$\displaystyle y \:=\:1- 1 \:=\:0$

The line intersects the parabola at $\displaystyle (1,0).$

Since the line intersects the parabola in one point,
. . the line is tangent to the parabola.

3. ## thx

thx for this i really apreciate it

but couldnt anyone solve the second one ???????