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

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

Four points A (6, 3), B (−3, 5), C (4, −2) and D (x, 3x) are given in such a way that \[\frac{\Delta DBC}{\Delta ABC} = \frac{1}{2}\]. Find x.

The points A (2, 0), B (9, 1), C (11, 6) and D (4, 4) are the vertices of a quadrilateral ABCD. Determine whether ABCD is a rhombus or not.

The base of an equilateral triangle with side 2a lies along the y-axis, such that the mid-point of the base is at the origin. Find the vertices of the triangle.