1. ## simultaneous equations and uadratic inequalities

1. find the values of k such that y=2x+k is a tangent to the curve with equation y=xsquared+1

2. find the values of k such that y=kx-2 is a tangent to the curve with equation y=xsquared+1

2. Hello, wale!

2. Find the values of $k$ such that $y\:=\:kx-2$
is a tangent to the curve with equation $y\:=\:x^2+1$

If the line is tangent to the parabola, they intersect at exactly one point.

Intersections: . $x^2+1 \:=\:kx-2 \quad\Rightarrow\quad x^2 - kx + 3 \:=\:0$

Quadratic Formula: . $x \;=\;\frac{k \pm \sqrt{k^2-12}}{2}$

If the quadratic equation has one root, the discriminant must be zero:

. . $k^2-12 \:=\:0 \quad\Rightarrow\quad k^2 \:=\:12 \quad\Rightarrow\quad k \:=\:\pm\sqrt{12} \:=\:\pm2\sqrt{3}$

3. Originally Posted by wale