Releted MCQ Question on
Geometry >> Hyperbola
Releted Question 1
Each of the four inequalities given below defines a region in the $$xy$$ plane. One of these four regions does not have the following property. For any two points $$\left( {{x_1},\,{y_1}} \right)$$ and $$\left( {{x_2},\,{y_2}} \right)$$ in the
the region, the point $$\left( {\frac{{{x_1} + {x_2}}}{2},\,\frac{{{y_1} + {y_2}}}{2}} \right)$$ is also in the region. The inequality defining this region is :
A.
$${x^2} + 2{y^2} \leqslant 1$$
B.
$${\text{max }}\left\{ {\left| x \right|,\left| y \right|} \right\} \leqslant 1$$
C.
$${x^2} - {y^2} \leqslant 1$$
D.
$${y^2} - {x^2} \leqslant 0$$
Releted Question 2
Let $$P\left( {a\,\sec \,\theta ,\,b\,\tan \,\theta } \right)$$ and $$Q\left( {a\,\sec \,\phi ,\,b\,\tan \,\phi } \right),$$ where $$\theta + \phi = \frac{\pi }{2},$$ be two points on the hyperbola $$\frac{{{x^2}}}{{{a^2}}} - \frac{{{y^2}}}{{{b^2}}} = 1.$$ If $$\left( {h,\,k} \right)$$ is the point of intersection of the normal at $$P$$ and $$Q,$$ then $$k$$ is equal to :
A.
$$\frac{{{a^2} + {b^2}}}{a}$$
B.
$$ - \left( {\frac{{{a^2} + {b^2}}}{a}} \right)$$
C.
$$\frac{{{a^2} + {b^2}}}{b}$$
D.
$$ - \left( {\frac{{{a^2} + {b^2}}}{b}} \right)$$
Releted Question 3
If $$x=9$$ is the chord of contact of the hyperbola $${x^2} - {y^2} = 9,$$ then the equation of the corresponding pair of tangents is :
A.
$$9{x^2} - 8{y^2} + 18x - 9 = 0$$
B.
$$9{x^2} - 8{y^2} - 18x + 9 = 0$$
C.
$$9{x^2} - 8{y^2} - 18x - 9 = 0$$
D.
$$9{x^2} - 8{y^2} + 18x + 9 = 0$$
Releted Question 4
For hyperbola $$\frac{{{x^2}}}{{{{\cos }^2}\alpha }} - \frac{{{y^2}}}{{{{\sin }^2}\alpha }} = 1,$$ which of the following remains constant with change in $$'\alpha \,'$$
A.
abscissae of vertices
B.
abscissae of foci
C.
eccentricity
D.
directrix
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Hyperbola