Hyperbola PYQs - Last 10 Years

The locus of the midpoint of the system of parallel chords parallel to the line \(y=2 x\) to the hyperbola \(9 x^2-4 y^2=36\) is

In a plane \(\vec{a}\) and \(\vec{b}\) are the position vectors of two points A and B respectively. A point $P$ with position vector \(\overrightarrow{\mathrm{r}}\) moves on that plane in such a way that $$|\overrightarrow{\vec{r}}-\vec{a}|...

Let \(A(2\sec \theta ,3\tan \theta )\) and \(B(2\sec \phi ,3\tan \phi )\) where \(\theta + \phi = {\pi \over 2}\) be two points on the hyperbola \({{{x^2}} \over 4} - {{{y^2}} \over 9} = 1\). If (\(\alpha,\beta\)) is the point of interse...

The average length of all vertical chords of the hyperbola \({{{x^2}} \over {{a^2}}} - {{{y^2}} \over {{b^2}}} = 1,a \le x \le 2a\), is :

If a hyperbola passes through the point P(\(\sqrt2\), \(\sqrt3\)) and has foci at (\(\pm\) 2, 0), then the tangent to this hyperbola at P is

The line y = x + 5 touches

PQ is a double ordinate of the hyperbola \({{{x^2}} \over {{a^2}}} - {{{y^2}} \over {{b^2}}} = 1\) such that \(\Delta OPQ\) is an equilateral triangle, O being the centre of the hyperbola. Then the eccentricity e of the hyperbola satisfies

Let \(P(3\sec \theta ,2\tan \theta )\) and \(Q(3\sec \phi ,2\tan \phi )\) be two points on \({{{x^2}} \over 9} - {{{y^2}} \over 4} = 1\) such that \(\theta + \phi = {\pi \over 2},0 < \theta ,\phi < {\pi \over 2}\). Then the ordinate of...

The locus of the centre of a variable circle which always touches two given circles externally is

The normal to a curve at P(x, y) meets the X-axis at G. If the distance of G from the origin is twice the abscissa of P then the curve is

A double ordinate PQ of the hyperbola \({{{x^2}} \over {{a^2}}} - {{{y^2}} \over {{b^2}}} = 1\) is such that \(\Delta OPQ\) is equilateral, O being the centre of the hyperbola. Then the eccentricity e satisfies the relation

For the hyperbola \({{{x^2}} \over {{{\cos }^2}\alpha }} - {{{y^2}} \over {{{\sin }^2}\alpha }} = 1\), which of the following remains fixed when \(\alpha\) varies?

A point is in motion along a hyperbola \(y = {{10} \over x}\) so that its abscissa x increases uniformly at a rate of 1 unit per second. Then, the rate of change of its ordinate when the point passes through (5, 2)

The equation of the directrices of the hyperbola \(3{x^2} - 3{y^2} - 18x + 12y + 2 = 0\) is

Equation of a tangent to the hyperbola 5x2 \(-\) y2 = 5 and which passes through an external point (2, 8) is

The length of conjugate axis of a hyperbola is greater than the length of transverse axis. Then, the eccentricity e is

Let P(4, 3) be a point on the hyperbola \({{{x^2}} \over {{a^2}}} - {{{y^2}} \over {{b^2}}} = 1\). If the normal at P intersects the X-axis at (16, 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 x2 + 9y2 = 9, then the ratio of a2 : b2 equals to

A hyperbola, having the transverse axis of length 2sin\(\theta\) is confocal wit6h the ellipse 3x2 + 4y2 = 12. Its equation is

Let A(\(-\) 1, 0) and B(2, 0) be two points. A point M moves in the plane in such a way that \(\angle MBA\) = 2\(\angle MAB\). Then, the point M moves along

Let P be the foot of the perpendicular from focus S of hyperbola \({{{x^2}} \over {{a^2}}} - {{{y^2}} \over {{b^2}}} = 1\) on the line bx \(-\) ay = 0 and let C be the centre of the hyperbola. Then, the area of the rectangle whose sides are...

The line segment joining the foci of the hyperbola \({x^2} - {y^2} + 1 = 0\) is one of the diameters of a circle. The equation of the circle is

If PQ is a double ordinate of the hyperbola \({{{x^2}} \over {{a^2}}} - {{{y^2}} \over {{b^2}}} = 1\) such that \(\Delta OPQ\) is equilateral. O being the centre. Then, the eccentricity e satisfies