Hyperbola

Consider the ellipse $E$ given by $\frac{x^2}{18}+\frac{y^2}{12}=1$. Let $H$ be the hyperbola whose eccentricity is the reciprocal of the eccentricity of $E$ and whose foci are the same as that of $E$. Let $P$ and $Q$ be the points of inter...

Match each entry in List-I to the correct entry in List-II and choose the correct option.






List-I

List-II











(P) The circle with centre $(1,2)$ and touching the straight line

...

Consider the hyperbola

\(\frac{x^{2}}{100}-\frac{y^{2}}{64}=1\)

with foci at $S$ and $S_{1}$, where $S$ lies on the positive $x$-axis. Let $P$ be a point on the hyperbola, in the first quadrant. Let $\angle S P S_{1}=\alpha$, with $\alp...

Let a and b be positive real numbers such that a > 1 and b < a. Let P be a point in the first quadrant that lies on the hyperbola \({{{x^2}} \over {{a^2}}} - {{{y^2}} \over {{b^2}}} = 1\). Suppose the tangent to the hyperbola at P passes th...

Let T be the line passing through the points P(\(-\)2, 7) and Q(2, \(-\)5). Let F1 be the set of al pairs of circles (S1, S2) such that T is tangent to S1 at P and tangent to S2 at Q, and also such that S1 and S2 touch each other at a point...

Let \(H:{{{x^2}} \over {{a^2}}} - {{{y^2}} \over {{b^2}}} = 1\), where a > b > 0, be a hyperbola in the XY-plane whose conjugate axis LM subtends an angle of 60\(^\circ\) at one of its vertices N. Let the area of the \(\Delta\)LMN be $$4\...

For \(a = \sqrt 2\), if a tangent is drawn to a suitable conic (Column 1) at the point of contact (\(-\)1, 1), then which of the following options is the only CORRECT combination for obtaining its equation?

The tangent to a suitable conic (Column 1) at \(\left( {\sqrt 3 ,\,{1 \over 2}} \right)\) is found to be \(\sqrt 3 x + 2y = 4\), then which of the following options is the only CORRECT combination?

If \(2x - y + 1 = 0\) is a tangent to the hyperbola \({{{x^2}} \over {{a^2}}} - {{{y^2}} \over {16}} = 1\) then which of the following CANNOT be sides of a right angled triangle?

Consider the hyperbola \(H:{x^2} - {y^2} = 1\) and a circle \(S\) with center \(N\left( {{x_2},0} \right)\). Suppose that \(H\) and \(S\) touch each other at a point \(P\left( {{x_1},{y_1}} \right)\) with \({{x_1} > 1}\) and \({{y_1} > 0}\)...

Tangents are drawn to the hyperbola \({{{x^2}} \over 9} - {{{y^2}} \over 4} = 1,\) parallel to the straight line \(2x - y = 1,\) The points of contact of the tangents on the hyperbola are

Let \(P(6, 3)\) be a point on the hyperbola \({{{x^2}} \over {{a^2}}} - {{{y^2}} \over {{b^2}}} = 1\). If the normal at the point \(P\) intersects the \(x\)-axis at \((9, 0)\), then the eccentricity of the hyperbola is

Let the eccentricity of the hyperbola \({{{x^2}} \over {{a^2}}} - {{{y^2}} \over {{b^2}}} = 1\) be reciprocal to that of the ellipse \({x^2} + 4{y^2} = 4\). If the hyperbola passes through a focus of the ellipse, then

The circle \({x^2} + {y^2} - 8x = 0\) and hyperbola \({{{x^2}} \over 9} - {{{y^2}} \over 4} = 1\) intersect at the points \(A\) and \(B\).
Equation of the circle with \(AB\) as its diameter is

The line \(2x + y = 1\) is tangent to the hyperbola \({{{x^2}} \over {{a^2}}} - {{{y^2}} \over {{b^2}}} = 1\). If this line passes through the point of intersection of the nearest directrix and the \(x\)-axis, then the eccentricity of the h...

The circle \({x^2} + {y^2} - 8x = 0\) and hyperbola \({{{x^2}} \over 9} - {{{y^2}} \over 4} = 1\) intersect at the points \(A\) and \(B\).
Equation of a common tangent with positive slope to the circle as well as to the hyperbola is

Consider a branch of the hyperbola
\(${x^2} - 2{y^2} - 2\sqrt 2 x - 4\sqrt 2 y - 6 = 0\)$
with vertex at the point \(A\). Let \(B\) be one of the end points of its latus rectum. If \(C\) is the focus of the hyperbola nearest to the point ...

A hyperbola, having the transverse axis of the length \(2\sin \theta\), is confocal with the ellipse \(3{x^2} + 4{y^2} = 12\). Then its equation is

A hyperbola, having the transverse axis of length \(2\sin \theta ,\) is confocal with the ellipse \(3{x^2} + 4{y^2} = 12.\) Then its equation is

Match the statements in Column \(I\) with the properties in Column \(II\) and indicate your answer by darkening the appropriate bubbles in the \(4 \times 4\) matrix given in the \(ORS\).
Column \(I\)
(A) Two intersecting circles
(B) Two m...

If a hyperbola passes through the focus of the ellipse $\frac{x^2}{25}+\frac{y^2}{16}=1$ and its transverse and conjugate axes coincide with the major and minor axes of the ellipse, and the product of eccentricities is 1 , then

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.

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.

If the line \(62x + \sqrt 6 y = 2\) touches the hyperbola \({x^2} - 2{y^2} = 4\), then the point of contact is

For hyperbola \({{{x^2}} \over {{{\cos }^2}\alpha }} - {{{y^2}} \over {{{\sin }^2}\alpha }} = 1\) which of the following remains constant with change in \('\alpha '\)

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

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...

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.

The equation \({{{x^2}} \over {1 - r}} - {{{y^2}} \over {1 + r}} = 1,\,\,\,\,r > 1\) represents

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 r...