Question

Let P(a cos ?, b sin ?), where ? is not a multiple of ?/2, be a point on the ellipse C : x^2/a^2 + y^2/b^2 = 1, where a ? b > 0; and P' (a cos ?, a sin ?) the corresponding point on the 'auxiliary circle' C' : x^2 + y^2 = a^2. Prove that the tangents at P to C and at P' to C' meet on the x-axis. Hint: Write down an affine transformation that maps C to C' and P to P', and that maps each point of the x-axis to itself.

Name: Let P(a cos ?, b sin ?), where ? is not a multiple of ?/2, be a point on the ellipse C : x^2/a^2 + y^2/b^2 = 1, where a ? b > 0; and P' (a cos ?, a sin ?) the corresponding point on the 'auxiliary circle' C' : x^2 + y^2 = a^2. Prove that the tangents at P to C and at P' to C' meet on the x-axis. Hint: Write down an affine transformation that maps C to C' and P to P', and that maps each point of the x-axis to itself.
Uploaded: 2023-01-12T07:11:49-08:00
Duration: 1 min 34 s
Channel: Sri K

          Let P(a cos ?, b sin ?), where ? is not a multiple of ?/2, be a point on the ellipse C : x^2/a^2 + y^2/b^2 = 1, where a ? b > 0; and P' (a cos ?, a sin ?) the corresponding point on the 'auxiliary circle' C' : x^2 + y^2 = a^2. Prove that the tangents at P to C and at P' to C' meet on the x-axis. Hint: Write down an affine transformation that maps C to C' and P to P', and that maps each point of the x-axis to itself.

Let P(a cos ?, b sin ?), where ? is not a multiple of ?/2, be a point on the ellipse C : x^2/a^2 + y^2/b^2 = 1, where a ? b > 0; and P' (a cos ?, a sin ?) the corresponding point on the 'auxiliary circle' C' : x^2 + y^2 = a^2. Prove that the tangents at P to C and at P' to C' meet on the x-axis. Hint: Write down an affine transformation that maps C to C' and P to P', and that maps each point of the x-axis to itself.

Added by Christopher D.

Calculus: Early Transcendentals

James Stewart 8th Edition

Instant Answer

Solved by Expert Sri K

01/12/2023

Step 1

First, let's find the inverse transformation that maps €C t0 € and P t0 p' : $$T(C,P) = \frac{1}{a} \left( \frac{P-P'}{a} \right)$$ Now, we can use this inverse transformation to find the corresponding points on C' : $$P'(a cos 0,a sin 0) = T(C',P)$$ Show more…

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Let P(a cos θ, b sin θ), where θ is not a multiple of π/2, be a point on the ellipse C: x^2/a^2 + y^2/b^2 = 1, where a > 0; and P' (a cos θ, a sin θ) the corresponding point on the "auxiliary circle" C': x^2 + y^2 = a^2. Prove that the tangents at P to C and at P' to C' meet on the x-axis. Hint: Write down an affine transformation that maps C to C' and P to P', and that maps each point on the x-axis to itself.