Class 12

Math

Co-ordinate Geometry

Ellipse

AB and CD are two equal and parallel chords of the ellipse $a_{2}x_{2} +b_{2}y_{2} =1$. Tangents to the ellipse at A and B intersect at P and tangents at C and D at Q. The line PQ

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A rod of length l cm moves with its ends A (on x=-axis) and B (on y-axis) always touching the coordinae axes. Prove that the point P on the rod which divides AB in the ratio $λ(=1)$ is ellipse. Alos, find the eccentricity of the ellipse.

An ellipse is drawn with the major and minor axes of lengths 10 and 8, rspectively. Using one focus as center a circle is drawn that is tangent ot the ellipse, with no part of the circle being outside the ellipse. Then find the radius of the circle.

If the tangent at any point of the ellipe e $a_{2}x_{2} +b_{2}y_{2} =1$ makes an angle $α$ with the major axis and angle $β$ with the focal radius of the point of contanct, then show that the eccentricity of the ellipse is given by $e=cosβ/α$.

Tangents are drawn from the points on the line x-y-5=0 ot $x_{2}+4y_{2}=4$ . Prove that all the chords of contanct pass through a fixed point

If the line $xcosα+ysinα=p$ is a tangent to the ellipse $a_{2}x_{2} +b_{2}y_{2} =1$, then prove that $a_{2}cos_{2}α+b_{2}sin_{2}α=p_{2}$

The length of the major axis of the ellipse $(5x−10)_{2}+(5y+15)_{2}=4(3x−4y+7)_{2} $ is

If a quadrilateral formed by four tangents to the ellipse $9x_{2} +4y_{2} =1$ is a square, then find the area of square.

Find the area between the ellpise $a_{2}x_{2} +b_{2}y_{2} =1$ and lines $a∣x∣ +b∣y∣ =1$