1. Coordinate Systems

Question 1:
A line with gradient 4 touches the parabola with equation y^2 = 4x. Find the coordinates of the point of contact of this tangent with the parabola and an equation of the normal to the parabola at that point.

Question 2:
The line with equation y = m(x+a), where ‘m’ can vary but ‘a’ is constant, meets the parabola with equation y^2 = 4ax in two points P and Q. Find, in terms of a and m, the coordinates of the mid-point R of PQ. As m varies show that R lies on the curve with equation y^2 = 2a(x+a).

2. Originally Posted by geton

Question 1:
A line with gradient 4 touches the parabola with equation y^2 = 4x. Find the coordinates of the point of contact of this tangent with the parabola and an equation of the normal to the parabola at that point.
taking the derivative implicitly we get

$\displaystyle 2y\frac{dy}{dx}=4$ but we know that $\displaystyle \frac{dy}{dx}=4$

so $\displaystyle 2y(4)=4 \iff y=\frac{1}{2}$ so $\displaystyle x=\frac{1}{16}$

The normal line will have slope $\displaystyle m=-\frac{1}{4}$

$\displaystyle y-\frac{1}{2}=-\frac{1}{4}(x-\frac{1}{16})$

$\displaystyle y=-\frac{1}{4}x+\frac{33}{64}$

3. Thank you so much TheEmptySet.

4. I've done my second problem myself.

Thank you.