   Chapter 10.1, Problem 84E

Chapter
Section
Textbook Problem

# Proof Prove Theorem 10.4 by showing that the tangent line to an ellipse at a point P makes equal angles with lines through P and the foci (see figure). [Hint: (1) Find the slope of the tangent line at P, (2) find the slopes of the lines through P and each focus, and (3) use the formula for the tangent of the angle between two lines.]

To determine

To prove: The theorem of reflective property of an ellipse which states, “If P is a point on an ellipse then the tangent line to the ellipse at point P makes equal angles with the lines through P and the foci.”.

Explanation

Given:

The figure,

And, the theorem, “If P is a point on an ellipse then the tangent line to the ellipse at point P makes equal angles with the lines through P and the foci.”.

Formula used:

The angle,α between the two lines whose slopes are m1 and m2 is given by,

α=tan1(|m1m21+m1m2|)

Proof:

Consider the given diagram,

Now, the equation of the ellipse is, x2a2+y2b2=1.

Now the equation, x2a2+y2b2=1 is equivalent to, b2x2+a2y2=a2b2.

Also in an ellipse, a2b2=c2.

Differentiate the curve, x2a2+y2b2=1 with respect to x.

So,

2xa2+2yyb2=0y=b2xa2y

Thus, the slope of the tangent line at any point on the curve is given by, b2xa2y.

The slope of the tangent at the point P(x0,y0) is given by, b2x0a2y0.

Now, the slope of the line passing through (c,0) and (x0,y0) is given by,

m1=y00x0(c)=y0x0+c

The slope of the line is, y0x0+c.

Now, use the formula, α=tan1(|m1m21+m1m2|) to find the angle between the tangent line and the line passing through P and (c,0).

So,

β=tan1(|b2x0a2y0y0x0+c1+(b2x0a2y0)(y0x0+c)|)=tan1(|b2x02b2cx0a2y02a2y0(x0+c)a2x0y0+a2cy0b2x0y0a2y0(x0+c)|)=tan1(|b2x02+b2cx0+a2y02a2x0y0+a2cy0b2x0y0|)=tan1(|b2x02+a2y02+b2cx0x0y0(a2b2)+a2cy0|)

Now, put b2x2+a2y2=a2b2 and a2b2=c2 in the expression, tan1(|b2x02+a2y02+b2cx0x0y0(a2b2)+a2cy0|)

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