Question #124559
Verify that the point P(acosθ,bsinθ) lies on the ellipse
x^2/ a^2+ y^2/ b^2= 1,
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 axsinθ−by cosθ = (a^2 −b^2)sinθ cosθ. If P is not on the axes and if the normal at P passes through the point B(0,b), Show that a^2 > 2b^2. If further, the tangent at P meets the y-axis at Q, show that
|BQ| =
a^2/ b^2
.
1
Expert's answer
2020-07-01T18:39:32-0400

equation of ellipse x2/a2 + y2/b2 = 1

we have points P(acosθ\theta + bsinθ\theta )

put the points in the equation of the ellipse

take L.H.S

(acosθ\theta)2/a2 + (bsinθ\theta)2/b2

a2cos2θ\theta /a2 + b2sin2 θ\theta/b2

cos2 θ\theta + sin2θ\theta

=1

hence the points are lies in the ellipse


the equation of normal is at p is ax sinθ\theta - by cosθ\theta = (a2-b2)sinθ\theta cosθ\theta

the gradient of the equtation is dy/dx

so the gradient is

dy/dx( ax sinθ\theta) - dy/dx(by cosθ\theta )= dy/dx {(a2-b2)sinθ\thetacosθ\theta}

asinθ\theta -0=0 is the gradient of the tangent to the curve at P


equation of the normal is

axsinθ\theta -bycosθ\theta = (a2-b2)sinθ\theta cosθ\theta at point B(0,b)

a(0)sinθ\theta - b(b)cosθ\theta = (a2-b2)sinθ\theta cosθ\theta

0-b2cosθ\theta =(a2-b2)sinθ\theta cosθ\theta

hence a2>2b2


if the tangent P meets the y axis at Q

than |BQ| = a2/b2

the point B (0,b) similarly at y axis the point Q (a,0)

so the the point BQ(a,b)

hence |BQ| = a2/b2

Need a fast expert's response?

Submit order

and get a quick answer at the best price

for any assignment or question with DETAILED EXPLANATIONS!

Comments

No comments. Be the first!
LATEST TUTORIALS
APPROVED BY CLIENTS