Class 11

Math

Co-ordinate Geometry

Conic Sections

A point P lies on the ellipe $64(y−1)_{2} +49(x+2)_{2} =1$ . If the distance of P from one focus is 10 units, then find its distance from other focus.

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Find the length of the chord of the ellipse $25x_{2} +16y_{2} =1$, whose middle point is $(21 ,52 )$.

Normal to the ellipse $64x_{2} +49y_{2} =1$ intersects the major and minor axes at $PandQ$ , respectively. Find the locus of the point dividing segment $PQ$ in the ratio 2:1.

If the normal at $P(2,233 )$ meets the major axis of ellipse $16x_{2} +9y_{2} =1$ at $Q$ , and $S$ and $S_{′}$ are the foci of the given ellipse, then find the ratio $SQ:S_{prime}Q˙$

$O$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.

If the tangent at any point of the ellipse $a_{3}x_{2} +b_{2}y_{2} =1$ makes an angle $α$ with the major axis and an angle $β$ with the focal radius of the point of contact, then show that the eccentricity of the ellipse is given by $e=cosαcosβ $

If the line $lx+my+n=0$ cuts the ellipse $(a_{2}x_{2} )+(b_{2}y_{2} )=1$ at points whose eccentric angles differ by $2π ,$ then find the value of $n_{2}a_{2}l_{2}+b_{2}m_{2} $ .

If the area of the ellipse $(a_{2}x_{2} )+(b_{2}y_{2} )=1$ is $4π$ , then find the maximum area of rectangle inscribed in the ellipse.

If $α−β=$ constant, then the locus of the point of intersection of tangents at $P(acosα,bsinα)$ and $Q(acosβ,bsinβ)$ to the ellipse $a_{2}x_{2} +b_{2}y_{2} =1$ is: (a) a circle (b) a straight line (c) an ellipse (d) a parabola