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Find the coordinates of the focus and the vertex, the equations of the directrix and the axis, and length of the latus rectum of the parabola :(i) x2 = 16y(ii) x2 = 10y(iii) 3x2 = 8y |
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Answer» The general form of a parabola: x2 = 4ay ….(1) Focus : F(0,a) Vertex : A(0,0) (at any point A) Equation of the directrix : y + a = 0 Axis: x = 0 Length of latus rectum : 4a (i) x2 = 16y On comparing given equation with (1), we have 4a = 16 => a = 4 Now, Focus : F(0, 4) Vertex : A(0, 0) Equation of the directrix : y + 4 = 0 Axis: x = 0 Length of latus rectum : 4a = 4 x 4 = 16 units (ii) x2 = 10y On comparing given equation with (1), we have 4a = 10 => a = 2.5 Now, Focus : F(0, 2.5) Vertex : A(0, 0) Equation of the directrix : y + 2.5 = 0 Axis: x = 0 Length of latus rectum : 4a = 4 x 2.5 = 10 units (iii) 3x2 = 8y or x2 = 8/3 y On comparing given equation with (1), we have 4a = 8/3 => a = 2/3 Now, Focus : F(0, 2/3) Vertex : A(0, 0) Equation of the directrix : y + 2/3 = 0 or 3y + 2 = 0 Axis: x = 0 Length of latus rectum : 4a = 4 x 2/3 = 8/3 units |
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