Class 12

Math

Co-ordinate Geometry

Conic Sections

A set of parallel chord of the parabola $y_{2}=4ax$ have their midpoint on

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Let $P$ be a point on the ellipse $a_{2}x_{2} +b_{2}y_{2} =1$ of eccentricity $e˙$ If $A,A_{′}$ are the vertices and $S,S$ are the foci of the ellipse, then find the ratio area $PSS_{′′}$ : area $APA_{prime}˙$

$O$is the origin & also the centre of two concentric circles having radii of the inner & the outer circle as \displaystyle{a}&{b} respectively. A line $OPQ$ is drawn to cut the inner circle in $P$ & the outer circle in $Q.PR$ is drawn parallel to the $y$-axis & $QR$ is drawn parallel to the $x$-axis. Prove that the locus of $R$ is an ellipse touching the two circles. If the focii of this ellipse lie on the inner circle, find the ratio of inner: outer radii & find also the eccentricity of the ellipse.

Find the equation for the ellipse that satisfies the given condition:

If (5, 12) and (24, 7) are the foci of an ellipse passing through the origin, then find the eccentricity of the ellipse.

Prove that the chord of contact of the ellipse $a_{2}x_{2} +b_{2}y_{2} =1$ with respect to any point on the directrix is a focal chord.

If $ω$ is one of the angles between the normals to the ellipse $a_{2}x_{2} +b_{2}y_{2} =1$ at the point whose eccentric angles are $θ$ and $2π +θ$ , then prove that $sin2θ2cotω =1−e_{2} e_{2} $

Find the eccentricity of an ellipse $a_{2}x_{2} +b_{2}y_{2} =1$ whose latus rectum is half of its major axis. $(a>b)$

Suppose that the foci of the ellipse $9x_{2} +5y_{2} =1$ are $(f_{1},0)and(f_{2},0)$ where $f_{1}>0andf_{2}<0.$ Let $P_{1}andP_{2}$ be two parabolas with a common vertex at (0, 0) and with foci at $(f_{1}.0)and$ (2f_2 , 0), respectively. Let$T_{1}$ be a tangent to $P_{1}$ which passes through $(2f_{2},0)$ and $T_{2}$ be a tangents to $P_{2}$ which passes through $(f_{1},0)$ . If $m_{1}$ is the slope of $T_{1}$ and $m_{2}$ is the slope of $T_{2},$ then the value of $(m121 +m22)$ is