Class 12

Math

Co-ordinate Geometry

Hyperbola

Consider a hyperbola $xy=4$ and a line $y=2x=4$. O is the centre of hyperbola. Tangent at any point P of hyperbola intersect the coordinate axes at A and B. Shortest distance between the line and hyperbola is

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If a hyperbola passing through the origin has $3x−4y−1=0$ and $4x−3y−6=0$ as its asymptotes, then find the equation of its transvers and conjugate axes.

The locus a point $P(α,β)$ moving under the condition that the line $y=αx+β$ is a tangent to the hyperbola $a_{2}x_{2} −b_{2}y_{2} =1$ is (A) a parabola (B) an ellipse (C) a hyperbola (D) a circle

Find the equation of hyperbola : Whose foci are (4, 2) and (8, 2) and accentricity is 2.

Statement 1 : Every line which cuts the hyperbola $4x_{2} −16y_{2} =1$ at two distinct points has slope lying in $(−2,2)˙$ Statement 2 : The slope of the tangents of a hyperbola lies in $(−∞,−2)∪(2,∞)˙$

The length of the transverse axis of the rectangular hyperbola $xy=18$ is 6 (b) 12 (c) 18 (d) 9

Find the eccentricity of the hyperbola given by equations $x=2e_{t}+e_{−1} andy=3e_{t}−e_{−1} ,t∈R˙$

Each of the four inequalities given below defines a region in the xy plane. One of these four regions does nothave the following property. For any two points $(x_{1},y_{2})and(y_{1},y_{2})$ in the region the piont $(2x_{1}+x_{2} ⋅2y_{1}+y_{2} )$ is also in the region. The inequality defining this region is

If $SandS_{′}$ are the foci, $C$ is the center, and $P$ is a point on a rectangular hyperbola, show that $SP×S_{prime}P=(CP)_{2}˙$