class 11

Math

Co-ordinate Geometry

Hyperbola

If $2x−y+1=0$ is a tangent to the hyperbola $a_{2}x_{2} −16y_{2} =1$ then which of the following CANNOT be sides of a right angled triangle? (a)$a,4,2$ (b) $a,4,1$(c)$2a,4,1$ (d) $2a,8,1$

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An ellipse has eccentricity $21 $ and one focus at the point $P(21 ,1)$. Its one directrix is the comionand tangent nearer to the point the P to the hyperbolaof $x_{2}−y_{2}=1$ and the circle $x_{2}+y_{2}=1$.Find the equation of the ellipse.

A variable chord of the hyperbola $a_{2}x_{2} −b_{2}y_{2} =1,(b>a),$ subtends a right angle at the center of the hyperbola if this chord touches. a fixed circle concentric with the hyperbola a fixed ellipse concentric with the hyperbola a fixed hyperbola concentric with the hyperbola a fixed parabola having vertex at (0, 0).

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

From the point (2, 2) tangent are drawn to the hyperbola $16x_{2} −9y_{2} =1.$ Then the point of contact lies in the first quadrant (b) second quadrant third quadrant (d) fourth quadrant

The distance between two directrices of a rectangular hyperbola is 10 units. Find the distance between its foci.

Find the value of $m$ for which $y=mx+6$ is tangent to the hyperbola $100x_{2} −49y_{2} =1$

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 a ray of light incident along the line $3x+(5−42 )y=15$ gets reflected from the hyperbola $16x_{2} −9y_{2} =1$ , then its reflected ray goes along the line. $x2 −y+5=0$ (b) $2 y−x+5=0$ $2 y−x−5=0$ (d) none of these