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For this problem on the topic of geometric optics, we are told that an elliptical mirror is described by the curve x squared over a squared plus y squared over b squared equal to 1.
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It has a semi -major axis a and semi -minor axis b.
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The foci of the ellipse are at c -0 and minus c -0 with c equal to a -square minus b squared all to the power half.
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You want to show that any light ray in the x -y plane which passes through one focus is reflected through the other.
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The two ray vectors here are defined as u, which is minus into x minus c x hat plus y y hat, and v, which is minus into x plus c x hat plus y y hat.
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They make angles of theta u and theta v with n respectively, and the normal vector to a surface is defined as n, which we can first determine.
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And n is df, d x, x hat, plus df, df, and we can write this as d by d x of x squared over a squared plus y squared over b squared x hat plus d, d ,y of x squared over a squared plus y squared over b squared y hat.
01:44
And so this becomes 2x over a squared x hat plus 2y over b squared y hat.
02:05
We have n equal to 2x over a squared x hat minus 2y over b squared y hat.
02:18
Now the unit vectors of u and v are given by u hat, which is the vector u divided by its magnitude, and this is equal to minus into x minus c x hat plus y y hat divided by the square root of x minus c all squared plus y squared.
02:49
The inner rector v hat is equal to v over the magnitude of v, which is minus into x plus c x hat plus y y hat divided by the square root of x plus c all squared plus y squared.
03:17
Now the dot product of n with two unit vectors, firstly n dot u hat is equal to the magnitude of n times the magnitude of u hat times cosine theta u, which is the magnitude of n cosine, which similarly n .v hat is the magnitude of n times cosine theta v...