class 11

Math

Co-ordinate Geometry

Conic Sections

Let P be the point on parabola $y_{2}=4x$ which is at the shortest distance from the center S of the circle $x_{2}+y_{2}−4x−16y+64=0$ let Q be the point on the circle dividing the line segment SP internally. Then

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Find the equation for the ellipse that given that satisfies the given conditions: Length of minor axis $16$, foci $(0,±6).$

Find the equation of the normal to the ellipse $a_{2}x_{2} +b_{2}y_{2} =1$ at the positive end of the latus rectum.

Find the coordinates of the focus, axis of the parabola ,the equation of directrix and the length of the latus reactum for $x$ $_{2} =6y$

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} $ .

Find the point $(α,β)$ on the ellipse $4x_{2}+3y_{2}=12,$ in the first quadrant, so that the area enclosed by the lines $y=x,y=β,x=α$ , and the x-axis is maximum.

A point $P$ moves such that the chord of contact of the pair of tangents from $P$ on the parabola $y_{2}=4ax$ touches the rectangular hyperbola $x_{2}−y_{2}=c_{2}˙$ Show that the locus of $P$ is the ellipse $c_{2}x_{2} +(2a)_{2}y_{2} =1.$

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

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) $tan_{−1}(a+b 2ab )$