# line of symmetry

• Jan 23rd 2013, 05:56 AM
jiboom
line of symmetry
having trouble with question 8 on this paper:

http://www.mei.org.uk/files/papers/2012_Jan_c3.pdf

im guessing its because the gradient at P is -1/3 and not -1 but i cant justify this. im thinking the gradient needs to be perpendicular to y=x bit can see no reason this helps :(
• Jan 23rd 2013, 07:18 AM
Deveno
Re: line of symmetry
along the line y = x we have to have dy/dx = -1 if the curve is symmetric, do we not?

think of it this way: suppose our curve WERE symmetric and $(x_1,y_1)$ is a point of the curve below y = x, and $(x_2,y_2)$ is the corresponding point of the curve above y = x.

if the curve is symmetric, we have $(x_2,y_2) = (y_1,x_1)$.

now, what is the slope between those two points? it's:

$\frac{y_2-y_1}{x_2-x_1} = \frac{x_1 - y_1}{y_1 - x_1} = -1$

now suppose a point where the curve crosses y = x is (a,a). then as $x_1 \to a$, the slope between the two points approaches dy/dx, but this slope is constantly -1.
• Jan 23rd 2013, 08:07 AM
jiboom
Re: line of symmetry
Quote:

Originally Posted by Deveno
along the line y = x we have to have dy/dx = -1 if the curve is symmetric, do we not?

think of it this way: suppose our curve WERE symmetric and $(x_1,y_1)$ is a point of the curve below y = x, and $(x_2,y_2)$ is the corresponding point of the curve above y = x.

if the curve is symmetric, we have $(x_2,y_2) = (y_1,x_1)$.

now, what is the slope between those two points? it's:

$\frac{y_2-y_1}{x_2-x_1} = \frac{x_1 - y_1}{y_1 - x_1} = -1$

now suppose a point where the curve crosses y = x is (a,a). then as $x_1 \to a$, the slope between the two points approaches dy/dx, but this slope is constantly -1.

hi,thanks for the reply. so its nothing to do with perpendicular? what would the grad need to be if the it was symmetric about y=kx for k not 1?
• Jan 23rd 2013, 12:22 PM
Deveno
Re: line of symmetry
well, yes, it does have to be perpendicular, but proving this for the line y = kx is a lot more troublesome (it's messy).

if a curve is symmetric about the line y = kx, then the point corresponding to $(x_1,y_1)$ below the line, is the point (above the line):

$(x_2,y_2) = \left(\frac{1-k^2}{1+k^2}x_1 + \frac{2k}{1+k^2}y_1,\ \frac{2k}{1+k^2}x_1 - \frac{1-k^2}{1+k^2}y_1\right)$

a similar (but more involved) calculation as before shows that:

$\frac{y_2-y_1}{x_2-x_1} = \frac{2kx_1-2y_1}{2ky_1-2k^2x_1} = \frac{1}{k}\cdot\frac{kx_1 - y_1}{y_1-kx_1} = -\frac{1}{k}$

which is perpendicular to the line.

intuitively, however, you can see this by rotating the line by arctan(k) - π/4