Class 12

Math

Co-ordinate Geometry

Hyperbola

Three points A,B and C taken on rectangular hyperbola $xy=4$ where $B(−2,−2)$ and $C(6,2/3)$. The normal at A is parallel to BC, then

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The lines parallel to the normal to the curve $xy=1$ is/are $3x+4y+5=0$ (b) $3x−4y+5=0$ $4x+3y+5=0$ (d) $3y−4x+5=0$

If the tangents to the parabola $y_{2}=4ax$ intersect the hyperbola $a_{2}x_{2} −b_{2}y_{2} =1$ at $AandB$ , then find the locus of the point of intersection of the tangents at $AandB˙$

An ellipse intersects the hyperbola $2x_{2}−2y=1$ orthogonally. The eccentricity of the ellipse is reciprocal to that of the hyperbola. If the axes of the ellipse are along the coordinate axes, then (b) the foci of ellipse are $(±1,0)$ (a) equation of ellipse is $x_{2}+2y_{2}=2$ (d) the foci of ellipse are $(t2,0)$ (c) equation of ellipse is $(x_{2}2y)$

$PQ$ and $RS$ are two perpendicular chords of the rectangular hyperbola $xy=c_{2}˙$ If $C$ is the center of the rectangular hyperbola, then find the value of product of the slopes of $CP,CQ,CR,$ and $CS˙$

The curve $xy=c,(c>0),$ and the circle $x_{2}+y_{2}=1$ touch at two points. Then the distance between the point of contacts is 1 (b) 2 (c) $22 $ (d) none of these

Find the lengths of the transvers and the conjugate axis, eccentricity, the coordinates of foci, vertices, the lengths of latus racta, and the equations of the directrices of the following hyperbola: $16x_{2}−9y_{2}=−144.$

The locus of the point which is such that the chord of contact of tangents drawn from it to the ellipse $a_{2}x_{2} +b_{2}y_{2} =1$ forms a triangle of constant area with the coordinate axes is a straight line (b) a hyperbola an ellipse (d) a circle

Find the equation of normal to the hyperbola $3x_{2}−y_{2}=1$ having slope $31 ˙$