class 11

Math

Co-ordinate Geometry

Hyperbola

A hyperbola passes through the point $P(2 ,3 )$and has foci at $(±2,0)˙$Then the tangent to this hyperbola at $P$also passes through the point :

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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 any line perpendicular to the transverse axis cuts the hyperbola $a_{2}x_{2} −b_{2}y_{2} =1$ and the conjugate hyperbola $a_{2}x_{2} −b_{2}y_{2} =−1$ at points $PandQ$ , respectively, then prove that normal at $PandQ$ meet on the x-axis.

A hyperbola is given with its vertices at $(−2,0)$ and $(2,0)$ one of the foci of hyperbola is $(3,0)$. Then which of the following points does not lie on hyperbola (A) $(24 ,5)$ (B) $(44 ,52 )$ (C) $(44 ,−52 )$ (D) $(−6,52 )$

The foci of the hyperbola $2x_{2}−3y_{2}=5$ are a.$(±56 ,0)$ b. $(±5/6,0)$ c. $(±5 /6,0)$ d. none of these

$OA$ and $OB$ are fixed straight lines, $P$ is any point and $PM$ and $PN$ are the perpendiculars from $P$ on $OAandOB,$ respectively. Find the locus of $P$ if the quadrilateral $OMPN$ is of constant area.

A tangent drawn to hyperbola $a_{2}x_{2} −b_{2}y_{2} =1$at $P(6π )$ forms a triangle of area $3a_{2}$ square units, with coordinate axes, then the squae of its eccentricity is equal to

Find the locus of the point of intersection of the lines $3x −y−43λ =0and3 λx+λy−43 =0$ for different values of $λ˙$

If $e_{1}ande_{2}$ are respectively the eccentricities of the ellipse $18x_{2} +4y_{2} =1$ and the hyperbola $9x_{2} −4y_{2} =1,$ then the relation between $e_{1}ande_{2}$ is a.$2e_{1}+e_{2}=3$ b. $e_{1}+2e_{2}=3$ c. $2e_{1}+e_{2}=3$ d. $e_{1}+3e_{2}=2$