PUC Science इयत्ता ११ - Karnataka Board PUC Question Bank Solutions for Mathematics

Find the centre, eccentricity, foci and directrice of the hyperbola .

16x² − 9y² + 32x + 36y − 164 = 0

Find the centre, eccentricity, foci and directrice of the hyperbola.

 x² − y² + 4x = 0

Find the centre, eccentricity, foci and directrice of the hyperbola .

x² − 3y² − 2x = 8.

Find the eccentricity of the hyperbola, the length of whose conjugate axis is \[\frac{3}{4}\] of the length of transverse axis.

If the distance between the foci of a hyperbola is 16 and its ecentricity is \[\sqrt{2}\],then obtain its equation.

Write the coordinates of the foci of the hyperbola 9x² − 16y² = 144.

Write the equation of the hyperbola of eccentricity \[\sqrt{2}\],  if it is known that the distance between its foci is 16.

If the foci of the ellipse \[\frac{x^2}{16} + \frac{y^2}{b^2} = 1\] and the hyperbola \[\frac{x^2}{144} - \frac{y^2}{81} = \frac{1}{25}\] coincide, write the value of b².

If e₁ and e₂ are respectively the eccentricities of the ellipse \[\frac{x^2}{18} + \frac{y^2}{4} = 1\]

and the hyperbola \[\frac{x^2}{9} - \frac{y^2}{4} = 1\] then write the value of 2 e₁² + e₂².

If e₁ and e₂ are respectively the eccentricities of the ellipse \[\frac{x^2}{18} + \frac{y^2}{4} = 1\] and the hyperbola \[\frac{x^2}{9} - \frac{y^2}{4} = 1\] , then the relation between e₁ and e₂ is

The equation of the conic with focus at (1, −1) directrix along x − y + 1 = 0 and eccentricity \[\sqrt{2}\] is

The eccentricity of the conic 9x² − 16y² = 144 is 

The eccentricity of the hyperbola whose latus-rectum is half of its transverse axis, is 

The eccentricity of the hyperbola x² − 4y² = 1 is 

The distance between the foci of a hyperbola is 16 and its eccentricity is \[\sqrt{2}\], then equation of the hyperbola is

If e₁ is the eccentricity of the conic 9x² + 4y² = 36 and e₂ is the eccentricity of the conic 9x² − 4y² = 36, then

The eccentricity the hyperbola \[x = \frac{a}{2}\left( t + \frac{1}{t} \right), y = \frac{a}{2}\left( t - \frac{1}{t} \right)\] is

The locus of the point of intersection of the lines \[\sqrt{3}x - y - 4\sqrt{3}\lambda = 0 \text { and } \sqrt{3}\lambda  + \lambda - 4\sqrt{3} = 0\]  is a hyperbola of eccentricity

If the line segment joining the points P (x₁, y₁) and Q (x₂, y₂) subtends an angle α at the origin O, prove that
OP · OQ cos α = x₁ x₂ + y₁, y₂