Class 12

Math

Co-ordinate Geometry

Conic Sections

Length of the latus rectum of the parabola $x +y =a $ is

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Find the coordinates of the foci, the vertices, the length of major axis, the minor axis, the eccentricity and the length of the latus rectum of the ellipse $100x_{2} +400y_{2} =1.$

Find the locus of the point which is such that the chord of contact of tangents drawn from it to the ellipse $a_{2}x_{2} +b_{2}y_{2} =1$ form a triangle of constant area with the coordinate axes.

Two perpendicular tangents drawn to the ellipse $25x_{2} +16y_{2} =1$ intersect on the curve.

If the normal at one end of the latus rectum of the ellipse $a_{2}x_{2} +b_{2}y_{2} =1$ passes through one end of the minor axis, then prove that eccentricity is constant.

Tangents are drawn to the ellipse $a_{2}x_{2} +b_{2}y_{2} =1,(a>b),$ and the circle $x_{2}+y_{2}=a_{2}$ at the points where a common ordinate cuts them (on the same side of the x-axis). Then the greatest acute angle between these tangents is given by (A) $tan_{−1}(2ab a−b )$ (B) $tan_{−1}(2ab a+b )$ (C) $tan_{−1}(a−b 2ab )$ (D) $tan_{−1}(a+b 2ab )$

If $F_{1}$ and $F_{2}$ are the feet of the perpendiculars from the foci $S_{1}andS_{2}$ of the ellipse $25x_{2} +16y_{2} =1$ on the tangent at any point $P$ on the ellipse, then prove that $S_{1}F_{1}+S_{2}F_{2}≥8.$

Tangents are drawn to the ellipse $a_{2}x_{2} +b_{2}y_{2} =1$ at two points whose eccentric angles are $α−β$ and $α+β$ The coordinates of their point of intersection are

A tangent is drawn to the ellipse to cut the ellipse $a_{2}x_{2} +b_{2}y_{2} =1$ and to cut the ellipse $c_{2}x_{2} +d_{2}y_{2} =1$ at the points P and Q. If the tangents are at right angles, then the value of $(c_{2}a_{2} )+(d_{2}b_{2} )$ is