If c > 1 / 2, how many lines through the point (0, c) are normal lines to the parabola y = x ^ 2? What if c <= 1/2? Could someone please help?

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- Apr 22nd 2010, 05:03 AMTsAmEParabola and normal lines
If c > 1 / 2, how many lines through the point (0, c) are normal lines to the parabola y = x ^ 2? What if c <= 1/2? Could someone please help?

- Apr 22nd 2010, 06:31 AMtom@ballooncalculus
Let a be the x-value of where the normal intersects x^2. Express m, the gradient of the normal line, in terms of a. Then (at least if you do a sketch) you can see that c, the height of where the normal intersects the y-axis, will be equal to a^2 minus ma, which you should find is equal to a^2 + 1/2. (Which is at least 1/2)

- Apr 22nd 2010, 02:52 PMTsAmE
I see but the answer was that there were 3 normal lines when c > 1/2 and 1 normal line when c <= 1/2.

- Apr 23rd 2010, 12:13 AMtom@ballooncalculus
- Apr 23rd 2010, 03:23 AMTsAmE
How could it be only 3 normal lines for c > 1/2? Aren't there infinite normal lines?

- Apr 23rd 2010, 04:38 AMtom@ballooncalculus
- Apr 23rd 2010, 07:34 AMTsAmE
- Apr 23rd 2010, 10:48 AMtom@ballooncalculus
You can have many lines (with different gradients) through a point (0,c) but only 3 of them will hit the curve at right-angles to it (or at right-angles to, rather, the curve's tangent at each of the points where they hit). Which is what it means for a line to be 'normal' to the curve.

- Apr 23rd 2010, 02:54 PMTsAmE
But couldnt the curve have many gradients as you arent given the point thats tangent to the curve? meaning the normal wont only cut 1 tangent at 90 degrees but will also cut the other tangents of different slopes? (y=2x gradient seems as if it could cut any where on the parabola)

- Apr 23rd 2010, 03:52 PMtom@ballooncalculus
A line is only a normal if it cuts the curve at 90 degrees, so any cuts it makes with tangents other than at 90 degrees

*and*on the curve are beside the point.

Good that you did the sketch, it will help you - but now make sure you only show lines that are truly (or at least roughly) normal to the curve.

Also, show a, and the geometry and/or algebra that enable you to determine c from it. E.g. the altitude of the point on the curve, a^2, and the height of a triangle with base a and hypotenuse with slope 1/(2a).