If the line [y = 4 + kx,,k 0], is the tangent to the parabola y = x - x^2 at the point P…

Question

If the line $$y = 4 + kx,\,k > 0$$, is the tangent to the parabola $y = x - {x^2}$ at the point P and V is the vertex of the parabola, then the slope of the line through P and V is :

Accepted Answer

$$\because$$ Line $y = kx + 4$ touches the parabola $y = x - {x^2}$.

So, $$kx + 4 = x - {x^2} \Rightarrow {x^2} + (k - 1)x + 4 = 0$$ has only one root

$${(k - 1)^2} = 16 \Rightarrow k = 5$$ or -3 but $k > 0$

So, $k = 5$.

And hence $${x^2} + 4x + 4 = 0 \Rightarrow x =  - 2$$

So, P(-2, -6) and V is $$\left( {{1 \over 2},{2 \over 4}} \right)$$

Slope of $$PV = {{{1 \over 4} + 6} \over {{1 \over 2} + 2}} = {5 \over 2}$$

Solutions · 1