Question #157500
Given that the equation of the tangent and normal to the ellipse (x^2/a^2) + (y^2/b^2) = 1 at the point P (acosu , bsinu) are respectively
bxcosu + aysinu = ab and
axsinu + bycosu = (a^2 - b^2)sinucosu.
Given that the tangents cuts the x - axis at A and the y - axis at B and that the normal cuts the x - axis at C and the y - axis at D. Show that as u varies, the locus of the midpoint of CD is
4[(a^2)*(x^2)]+ 4[(b^2)*(y^2)] = (a^2 - b^2)^2
Given that a = 5 and b = 4, sketch this locus
1
Expert's answer
2021-02-02T06:09:26-0500

Point C: y=0    ax=(a2b2)cosuy=0\implies ax=(a^2-b^2)cosu

Point D: x=0    by=(a2b2)sinux=0\implies by=(a^2-b^2)sinu

Midpoint of CD: xM=(a2b2)cosu2a,yM=(a2b2)sinu2bx_M=\frac{(a^2-b^2)cosu}{2a}, y_M=\frac{(a^2-b^2)sinu}{2b}

cos2u+sin2u=1cos^2u+sin^2u=1

4a2xM2(a2b2)2+4b2yM2(a2b2)2=1\frac{4a^2x_M^2}{(a^2-b^2)^2}+\frac{4b^2y_M^2}{(a^2-b^2)^2}=1


4a2xM2+4b2yM2=(a2b2)24a^2x_M^2+4b^2y_M^2=(a^2-b^2)^2





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