2005
JEE Advanced
Numerical
IIT-JEE 2005
Tangents are drawn from any point on the hyperbola ${{{x^2}} \over 9} - {{{y^2}} \over 4} = 1$ to the circle ${x^2} + {y^2} = 9$.Find the locus of mid-point of the chord of contact.
Correct Answer: $${{{x^2}} \over 9} - {{{y^2}} \over 4} = {\left( {{{{x^2} + {y^2}} \over 9}} \right)^2}$$
2005
JEE Advanced
MCQ
IIT-JEE 2005 Mains
Tangents are drawn from any point on the hyperbola $\frac{x^{2}}{9}-\frac{y^{2}}{4}=1$ to the circle $x^{2}+y^{2}=9$. Find the locus of mid-point of the chord of contact.
A.
${{{x^2}} \over 4} + {{{y^2}} \over 9} = {{{{({x^2} + {y^2})}^2}} \over {81}}$
B.
${{{x^2}} \over 4} - {{{y^2}} \over 9} = {{{{({x^2} + {y^2})}^2}} \over {81}}$
C.
${{{x^2}} \over 9} + {{{y^2}} \over 4} = {{{{({x^2} + {y^2})}^2}} \over {81}}$
D.
${{{x^2}} \over 9} - {{{y^2}} \over 4} = {{{{({x^2} + {y^2})}^2}} \over {81}}$
2004
JEE Advanced
MCQ
IIT-JEE 2004 Screening
If the line $62x + \sqrt 6 y = 2$ touches the hyperbola ${x^2} - 2{y^2} = 4$, then the point of contact is
A.
$\left( { - 2,\,\sqrt 6 } \right)$
B.
$\left( { - 5,\,2\sqrt 6 } \right)$
C.
$\left( {{1 \over 2},{1 \over {\sqrt 6 }}} \right)$
D.
$\left( {4, - \,\sqrt 6 } \right)$
2003
JEE Mains
MCQ
AIEEE 2003
The foci of the ellipse ${{{x^2}} \over {16}} + {{{y^2}} \over {{b^2}}} = 1$ and the hyperbola ${{{x^2}} \over {144}} - {{{y^2}} \over {81}} = {1 \over {25}}$ coincide. Then the value of ${b^2}$ is :
A.
$9$
B.
$1$
C.
$5$
D.
$7$
2003
JEE Advanced
MCQ
IIT-JEE 2003 Screening
For hyperbola ${{{x^2}} \over {{{\cos }^2}\alpha }} - {{{y^2}} \over {{{\sin }^2}\alpha }} = 1$ which of the following remains constant with change in $'\alpha '$
A.
abscissae of vertices
B.
abscissae of foci
C.
eccentricity
D.
directrix
1999
JEE Advanced
MCQ
IIT-JEE 1999
Let $P$ $\left( {a\,\sec \,\theta ,\,\,b\,\tan \theta } \right)$ and $Q$ $\left( {a\,\sec \,\,\phi ,\,\,b\,\tan \,\phi } \right)$, where $\theta + \phi = \pi /2,$, be two points on the hyperbola ${{{x^2}} \over {{a^2}}} - {{{y^2}} \over {{b^2}}} = 1$.
If $(h, k)$ is the point of intersection of the normals at $P$ and $Q$, then $k$ is equal to
A.
${{{a^2} + {b^2}} \over a}$
B.
$ - \left( {{{{a^2} + {b^2}} \over a}} \right)$
C.
${{{a^2} + {b^2}} \over b}$
D.
$ - \left( {{{{a^2} + {b^2}} \over b}} \right)$
1999
JEE Advanced
MCQ
IIT-JEE 1999
If $x$ $=$ $9$ is the chord of contact of the hyperbola ${x^2} - {y^2} = 9,$ then the equation of the vcorresponding 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.
$9{x^2} - 8{y^2} + 18x + 9 = 0$
1998
JEE Advanced
Numerical
IIT-JEE 1998
The angle between a pair of tangents drawn from a point $P$ to the parabola ${y^2} = 4ax$ is ${45^ \circ }$. Show that the locus of the point $P$ is a hyperbola.
Correct Answer: Solve it.
1981
JEE Advanced
MCQ
IIT-JEE 1981
The equation ${{{x^2}} \over {1 - r}} - {{{y^2}} \over {1 + r}} = 1,\,\,\,\,r > 1$ represents
A.
an ellipse
B.
a hyperbola
C.
a circle
D.
none of these
1981
JEE Advanced
MCQ
IIT-JEE 1981
Each of the four inequalties 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 region, the point $\left( {{{{x_1} + {x_2}} \over 2},{{{y_1} + {y_2}} \over 2}} \right)$ is also in the region. The inequality defining this region is
A.
${x^2} + 2{y^2} \le 1$
B.
Max $\left\{ {\left| x \right|,\left| y \right|} \right\} \le 1$
C.
${x^2} - {y^2} \le 1$
D.
${y^2} - x \le 0$