# [SOLVED] Error in this tangent line problem?

• Apr 24th 2009, 12:44 PM
sinewave85
[SOLVED] Error in this tangent line problem?
This is from a review section, and I am wondering whether there is a typo in the original problem itself or if I have forgotten something critical. Here is the problem:

"Find the point(s) on the graph of $\displaystyle f(x) = -x^2$ such that the tangent line at that point passes through the point (0,-9)."

Just thinking about the graph of $\displaystyle f(x) = -x^2$, it would seem impossible for there to be a tangent which passes throught (0,-9). But just for good measure, I did the math:

$\displaystyle \mbox{Let } f(x) = f(c) \mbox{ and } (x,y) = (c,-c^2)$

$\displaystyle m = f^{\prime}(c) = -2c$

$\displaystyle m = \frac{y_{2} - y_{1}}{x_{2} - x_{1}}$

$\displaystyle m = \frac{-9 + c^2}{0 - c} = \frac{-9 + c^2}{-c}$

$\displaystyle -2c = \frac{-9 + c}{-c}$

$\displaystyle 2c^2 = -9 + c^2$

$\displaystyle c^2 = -9$

$\displaystyle c = \sqrt{-9}$

Sure enough, there appears to be no real answer. Am I making a mistake, or is this a question without an answer?
• Apr 24th 2009, 12:48 PM
redsoxfan325
Quote:

Originally Posted by sinewave85
This is from a review section, and I am wondering whether there is a typo in the original problem itself or if I have forgotten something critical. Here is the problem:

"Find the point(s) on the graph of $\displaystyle f(x) = -x^2$ such that the tangent line at that point passes through the point (0,-9)."

Just thinking about the graph of $\displaystyle f(x) = -x^2$, it would seem impossible for there to be a tangent which passes throught (0,-9). But just for good measure, I did the math:

$\displaystyle \mbox{Let } f(x) = f(c) \mbox{ and } (x,y) = (c,-c^2)$

$\displaystyle m = f^{\prime}(c) = -2c$

$\displaystyle m = \frac{y_{2} - y_{1}}{x_{2} - x_{1}}$

$\displaystyle m = \frac{-9 + c^2}{0 - c} = \frac{-9 + c^2}{-c}$

$\displaystyle -2c = \frac{-9 + c}{-c}$

$\displaystyle 2c^2 = -9 + c^2$

$\displaystyle c^2 = -9$

$\displaystyle c = \sqrt{-9}$

Sure enough, there appears to be no real answer. Am I making a mistake, or is this a question without an answer?

I think you're right. The point $\displaystyle (0,-9)$ lies inside the parabola, so therefore there can be no tangent line to the parabola passing through this point.
• Apr 24th 2009, 12:54 PM
sinewave85
Quote:

Originally Posted by redsoxfan325
I think you're right. The point $\displaystyle (0,-9)$ lies inside the parabola, so therefore there can be no tangent line to the parabola passing through this point.

Thanks for the reinforcement, redsoxfan325. I guess I will just submit the problem as I have it worked up here. There is the possibility that the problem was purposefully designed to have no answer, but I think that it is more likely that the function and the point were supposed to have opposite signs.