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QUESTION: Let C:y=f(x)=ax^2+bx+c, xER, where a, b, and c are real constants with a different from 0. Prove that if ac>0, then there are two tangent lines to C that pass through the origin, but if ac<0 then no tangent line to C passes through the origin. What happens if ac=0?
ANSWER: If ac=0, either a=0 or c=0.
If a=0, f(x) is a line bx + c.
If c=0, f(x) = ax^2 + bx = x(ax+b) and at x=0, f(x) = 0.
---------- FOLLOW-UP ----------
QUESTION: How does your answer prove that there are two tangents lines to the graph C through the origin when ac>0, and no tangent line when ac<0. Please see the pdf file that I attached.
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I thought about the problem for so long that I forgot that we had to prove there were two tangent lines, but thanks for the response asking for it.
All parabolas have all slopes from -infinity to +infinity since the derivative of
ax^2 + bx + c is 2ax + b, and since x goes over all reals, the slope does as well.
The only way a tangent line could not be found to pass through that point was if the origin were inside the parabola.
Let d be negative and e be positve roots of the equation.
The parabola is then (x-d)(x-e), which multiplies to x^2 - (d+e)x + ed.
Note that if d is negative and e is positive, ed is negative, and that doesn't match the positive
coefficient on the x^2. To make the parabola open downwards, multiply (x-d)(x-e) by a negative.
In this case, x^2 has a negative coefficient, but since d and e have opposite signs, they're
product is minus a negative and is therefore positive. This still gives opposite signs.
From this, it can be seen that if there are root to the equation, they are both negative or
both positive. That says that the origin is outside the parabola, and and point on the outside
has two tangent lines to the parabola.
Just think about a line draw from that point. Since the point is outside the parabola,
a line could be constructed with a slope so that it wouldn't touch the parabola.
Just start modifying the slope until it touches the parabola at only one point.
This is the tangent line to the curve. This can be done on both sides of the parabola,
so there are two lines like this.
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