Class 11

Math

Co-ordinate Geometry

Hyperbola

Find the eccentricity, coordinates of the foci, equations of directrices and length of the latus rectum of the hyperbola $16x_{2}−9y_{2}=144$

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If the distance between the foci and the distance between the two directricies of the hyperbola $a_{2}x_{2} −b_{2}y_{2} =1$ are in the ratio 3:2, then $b:a$ is (a)$1:2 $ (b) $3 :2 $ (c)$1:2$ (d) $2:1$

From the center $C$ of hyperbola $a_{2}x_{2} −b_{2}y_{2} =1$ , perpendicular $CN$ is drawn on any tangent to it at the point $P(asecθ,btanθ)$ in the first quadrant. Find the value of $θ$ so that the area of $CPN$ is maximum.

If tangents drawn from the point $(a,2)$ to the hyperbola $16x_{2} −9y_{2} =1$ are perpendicular, then the value of $a_{2}$ is _____

The equation of conjugate axis of the hyperbola $xy−3y−4x+7=0$ is $y+x=3$ (b) $y+x=7$ $y−x=3$ (d) none of these

The equation of one of the directrices of a hyperboda is $2x+y=1,$ the corresponding focus is (1, 2) and $e=3 $ . Find the equation of the hyperbola and the coordinates of the center and the second focus.

A variable line $y=mx−1$ cuts the lines $x=2y$ and $y=−2x$ at points $AandB$ . Prove that the locus of the centroid of triangle $OAB(O$ being the origin) is a hyperbola passing through the origin.

If two distinct tangents can be drawn from the Point $(α,2)$ on different branches of the hyperbola $9x_{2} −16y_{2} =1$ then (1) $∣α∣<23 $ (2) $∣α∣>32 $ (3)$∣α∣>3$ (4) $α=1$

$C$ is the center of the hyperbola $a_{2}x_{2} −b_{2}y_{2} =1$ The tangent at any point $P$ on this hyperbola meet the straight lines $bx−ay=0$ and $bx+ay=0$ at points $QandR$ , respectively. Then prove that $CQC˙R=a_{2}+b_{2}˙$