Verify that the point P(a cos 0, b sin 0) lies on the ellipse y? = 1 62 a2 where a and b are the semi-major and semi-minor axes respectively of the ellipse . Find the gradient of the tangent to the curve at P and show that the equation of the normal at P is ax sin 0 – by cos 0 = (a² – b?) sin 0 cos 0. If P is not on the axes and if the normal at P passes through the point B(0,b), Show that a? > 262. If further, the tangent at P meets the y-axis at Q, show that a² |BQ| = 62 %3D

Elementary Geometry For College Students, 7e
7th Edition
ISBN:9781337614085
Author:Alexander, Daniel C.; Koeberlein, Geralyn M.
Publisher:Alexander, Daniel C.; Koeberlein, Geralyn M.
ChapterP: Preliminary Concepts
SectionP.CT: Test
Problem 1CT
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Verify that the point P(a cos 0, b sin 0) lies on the ellipse
y?
= 1
62
a2
where a and b are the semi-major and semi-minor axes respectively of the ellipse . Find the
gradient of the tangent to the curve at P and show that the equation of the normal at P is
ax sin 0 – by cos 0 = (a² – b?) sin 0 cos 0.
If P is not on the axes and if the normal at P passes through the point B(0,b), Show that
a? > 262. If further, the tangent at P meets the y-axis at Q, show that
a²
|BQ| =
62
%3D
Transcribed Image Text:Verify that the point P(a cos 0, b sin 0) lies on the ellipse y? = 1 62 a2 where a and b are the semi-major and semi-minor axes respectively of the ellipse . Find the gradient of the tangent to the curve at P and show that the equation of the normal at P is ax sin 0 – by cos 0 = (a² – b?) sin 0 cos 0. If P is not on the axes and if the normal at P passes through the point B(0,b), Show that a? > 262. If further, the tangent at P meets the y-axis at Q, show that a² |BQ| = 62 %3D
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a) Verification if P(acosθ,bsinθ) lies on the ellipse

Geometry homework question answer, step 2, image 1

b) Finding the gradient of the tangent at point P

Geometry homework question answer, step 3, image 1

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