Question

Let the function f(x) = 2x2 – logex, x> 0, be decreasing in (0, a) and increasing in (a, 4). A tangent to the parabola y2 = 4ax at a point P on it passes through the point (8a, 8a –1) but does not pass through the point (-1/a, 0). If the equation of the normal at P is
$\frac{x}{α}+\frac{y}{β}=1$
then α + β is equal to _______ .

Answer 1

The correct answer is 45
$δ^′(x)=\frac{4x^2−1}{x}$
so f(x) is decreasing in $(0,\frac{1}{2})$ and increasing in $(\frac{1}{2},∞)$
$⇒a=\frac{1}{2}$
Tangent at $y^2=2x$
$⇒y=mx+\frac{1}{2m}$
It is passing through (4, 3)
$3=4m+\frac{1}{2m}$
$⇒m=\frac{1}{2} or \frac{1}{4}$
So tangent may be
$y=\frac{1}{2}x+1 or\ y=\frac{1}{4}x+2$
But $y=\frac{1}{2}x+1$ passes through (–2, 0) so rejected.
Equation of Normal
$y=−4x−2(\frac{1}{2})(−4)−\frac{1}{2}(−4)^3$
$y=−4x+4+32$
$\frac{x}{9}+\frac{y}{36}=1$
α + β = 9 + 36
= 45