Geometry

Conic Sections

Conics - Ellipse - General

         

Let (m,n)(m,n) be the point at which a line is tangent to the ellipse x216+y2100=1.\displaystyle{\frac{x^2}{16}+\frac{y^2}{100}=1}. If this line has xx-intercept A=(a,0)A=(a,0) and yy-intercept B=(0,b),B=(0,b), where aa and bb are positive, what is the minimum value of ab?ab?

The ellipse x28+y217=1\displaystyle{\frac{x^2}{8}+\frac{y^2}{17}=1} and the line y=2xky=2x - k intersect at exactly two distinct points. What is the range of the constant k?k?

Find the equation of the tangent line to the ellipse 1x2+y2=531x^2+y^2=53 at the point (2,7)(2,7) on the ellipse.

In the above diagram, a rectangle ABCDABCD is inscribed in the ellipse x236+y2100=1.\displaystyle{\frac{x^2}{36}+\frac{y^2}{100}=1}. What is the maximum area of rectangle ABCD? ABCD?

Note: The above diagram is not drawn to scale.

Find the maximum area of the rectangle which is inscribed in the ellipse x252+y222=1.\frac{x^{2}}{5^{2}}+\frac{y^{2}}{2^{2}} = 1.

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