0
$\begingroup$

The line l:

  • Intersects the line $f(x) = \sqrt{x^2 +9} - x^2 + 5x$

  • Is parallel to the line $m: y = 5x-2 $

Find the formula of line l algebraically.

So, I found the derivative of f, which is $ \dfrac{x} {\sqrt{x^2+9}} - 2x+5$, and I know that line l has a slope of 5. So I got to the point $ \dfrac{x} {\sqrt{x^2+9}} - 2x = 0 $, but I don't know how to solve this..

  • 0
    Subtract 5 from both sides.2012-10-09
  • 0
    Yes, but after that you still have something I can't solve.2012-10-09
  • 0
    Add $2x$ to both sides, and then square both sides.2012-10-09
  • 0
    To be honest, I think something is up with the question. The only line that is tangent to $m$ is $m$, so that's your answer. It would make a lot more sense if it was tangent to $f$ and intersected $m$ (which seems to be what the question was, given your work).2012-10-09
  • 0
    I made a mistake, parallel to, not tangent to, sorry!2012-10-09
  • 1
    There are many lines that intersect the graph of $f$ and are parallel to the line $m$. ($f$ is not a line.) The line $m$ itself intersects $f$.2012-10-09

1 Answers 1

0

You have $\frac{x}{\sqrt{x^2+9}} - 2x = x(\frac{1}{\sqrt{x^2+9}}-2)$. So either $x = 0$ or $\frac{1}{\sqrt{x^2+9}}=2$ or $\frac{1}{x^2+9}=4$ or $4(x^2+9) = 1$. This one you should be able to solve.

  • 0
    Why $4(x^2+9)=1$? Where does that come from?2012-10-09