(1) Find the equation of the parabola in each of the cases given below:
(i) focus (4, 0) and directrix x = −4. Solution
(ii) passes through (2,-3) and symmetric about y -axis. Solution
(iii) vertex (1,-2) and focus (4,-2). Solution
(iv) end points of latus rectum(4,-8) and (4,8) . Solution
(2) Find the equation of the ellipse in each of the cases given below:
(i) foci (± 3 0), e = 1/2 Solution
(ii) foci (0, ± 4) and end points of major axis are (0, ± 5). Solution
(iii) length of latus rectum 8, eccentricity = 3/5 and major axis on x -axis. Solution
(iv) length of latus rectum 4 , distance between foci 4√2 and major axis as y - axis. Solution
(3) Find the equation of the hyperbola in each of the cases given below:
(i) foci (± 2, 0) , eccentricity = 3/2. Solution
(ii) Centre (2, 1) , one of the foci (8, 1) and corresponding directrix x = 4 . Solution
(iii) passing through (5,−2) and length of the transverse axis along x axis and of length 8 units. Solution
(4) Find the vertex, focus, equation of directrix and length of the latus rectum of the following:
(i) y^{2} = 16x Solution
(ii) x^{2} = 24y Solution
(iii) y^{2} = −8x Solution
(iv) x^{2} - 2x + 8y + 17 = 0 Solution
(v) y^{2} - 4y - 8x + 12 = 0 Solution
(5) Prove that the length of the latus rectum of the hyperbola (x^{2}/a^{2}) - (y^{2}/b^{2}) = 1 is 2b^{2}/a. Solution
(6) Show that the absolute value of difference of the focal distances of any point P on the hyperbola is the length of its transverse axis Solution
(7) Identify the type of conic and find centre, foci, vertices, and directrices of each of the following:
(i) (x^{2}/25) + (y^{2}/9) = 1 Solution
(ii) (x^{2}/3) + (y^{2}/10) = 1 Solution
(iii) (x^{2}/25) - (y^{2}/144) = 1 Solution
(iv) (y^{2}/16) - (x^{2}/9) = 1 Solution
(8) Identify the type of conic and find centre, foci, vertices, and directrices of each of the following:
(i) [(x - 3)^{2}/225] + [(y - 4)^{2}/289] = 1 Solution
(ii) [(x + 1)^{2}/100] + [(y - 2)^{2}/64] = 1 Solution
(iii) [(x + 3)^{2}/225] - [(y - 4)^{2}/64] = 1 Solution
(iv) [(y - 2)^{2}/25] - [(x + 1)^{2}/16] = 1 Solution
(v) 18x^{2} + 12y^{2} − 144x + 48y + 120 = 0 Solution
(vi) 9x^{2} − y^{2} − 36x − 6y + 18 = 0 Solution
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