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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.
Concept: undefined >> undefined
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
Concept: undefined >> undefined
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The equation of the conic with focus at (1, −1) directrix along x − y + 1 = 0 and eccentricity \[\sqrt{2}\] is
Concept: undefined >> undefined
The eccentricity of the conic 9x2 − 16y2 = 144 is
Concept: undefined >> undefined
The eccentricity of the hyperbola whose latus-rectum is half of its transverse axis, is
Concept: undefined >> undefined
The eccentricity of the hyperbola x2 − 4y2 = 1 is
Concept: undefined >> undefined
The distance between the foci of a hyperbola is 16 and its eccentricity is \[\sqrt{2}\], then equation of the hyperbola is
Concept: undefined >> undefined
If e1 is the eccentricity of the conic 9x2 + 4y2 = 36 and e2 is the eccentricity of the conic 9x2 − 4y2 = 36, then
Concept: undefined >> undefined
If the eccentricity of the hyperbola x2 − y2 sec2α = 5 is \[\sqrt{3}\] times the eccentricity of the ellipse x2 sec2 α + y2 = 25, then α =
Concept: undefined >> undefined
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
Concept: undefined >> undefined
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
Concept: undefined >> undefined
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
Concept: undefined >> undefined
The vertices of a triangle ABC are A (0, 0), B (2, −1) and C (9, 2). Find cos B.
Concept: undefined >> undefined
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.
Concept: undefined >> undefined
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.
Concept: undefined >> undefined
Find the coordinates of the centre of the circle inscribed in a triangle whose vertices are (−36, 7), (20, 7) and (0, −8).
Concept: undefined >> undefined
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.
Concept: undefined >> undefined
Find the distance between P (x1, y1) and Q (x2, y2) when (i) PQ is parallel to the y-axis (ii) PQ is parallel to the x-axis.
Concept: undefined >> undefined
Find a point on the x-axis, which is equidistant from the points (7, 6) and (3, 4).
Concept: undefined >> undefined
Find the locus of a point equidistant from the point (2, 4) and the y-axis.
Concept: undefined >> undefined
