Area of triangle and length of a curve

Question

From the point $(0,0)$ have been set two tangent lines towards the graph of the function $$\ y= x^2 + 8x + 9. $$

Find:

a) The area of the triangle which vertices are the points $(0,0)$ and the points at which the tangent lines touch the graph of the function

b) The length of the curve from the first to the second points where the tangent lines touch the graph of the function

I just can not figure out where the tangent lines touch the function, which seems to be a critical point in solving this problem.

Answer 1

HINT:

The equation of any line passing through $(0,0): y=mx$

Let us find the intersection: $$mx=x^2+8x+9\iff x^2+(8-m)x+9=0$$ which is a quadratic equation in $x,$ each value represents the abscissa of the intersection.

For tangency, the roots must be same.

Can you take it from here?

Answer 2

$y=x^2+8x+9\iff(x+4)^2=y+7$

WLOG any point on the parabola will be $(m-4,m^2-7)$

Now the gradient at $(m-4,m^2-7)$ will be $2(m-4)+8=1m$

So, the equation of the tangent will be $$\dfrac{y-(m^2-7)}{x-(m-4)}=2m$$

Now use the fact: the tangent has to pass through $(0,0)$ which will give us the two unequal values of $m$