\\(x^2\\over a^2\\) + \\(y^2\\over b^2\\) = 1,<\/p>\n
where a > b & \\(b^2\\) = \\(a^2(1 – e^2)\\) \\(\\implies\\) \\(a^2\\) – \\(b^2\\) = \\(a^2e^2\\).<\/p>\n<\/blockquote>\n
where e = eccentricity (0 < e < 1)<\/p>\n
Foci<\/strong> : S = (ae, 0) & S’ = (-ae, 0)<\/p>\n Vertices<\/strong> : A’ = (-a, 0) and\u00a0 A’ = (a, 0)<\/p>\n x = \\(a\\over e\\)\u00a0 and\u00a0 x = \\(-a\\over e\\)<\/p>\n The line segment A’A in which the foci S’ & S lie is of length 2a & is called the major axis (a > b) of the ellipse. The Point of intersection of major axis with directrix is called the foot of the directrix(z).<\/p>\n The y-axis intersects the ellipse in the points B’ = (0,-b) & B = (0,b). The line segment B’B of length 2b (b < a) is called the minor axis of the ellipse.<\/p>\n Both the axes minor and major together are called Principal Axes of the ellipse.<\/p>\n A chord perpendicular to major axis is called double ordinate of ellipse.<\/p>\n The focal chord perpendicular to major axis is called the latus rectum of ellipse.<\/p>\n (i)\u00a0 Length of latus rectum(LL’) = \\(2b^2\\over a\\) = \\({(minor axis)}^2\\over {major axis}\\) = 2a(1 – \\(e^2\\))<\/p>\n (ii) Equation of latus rectum : x = \\(\\pm\\)ae<\/p>\n (iii)\u00a0 Ends of latus rectum are L(ae, \\(b^2\\over a\\)), L'(ae, -\\(b^2\\over a\\)), L1(-ae, \\(b^2\\over a\\)), e = \\(\\sqrt{1 – {b^2\\over a^2}}\\)<\/p>\n\n\n Example : <\/span> Find the equation of ellipse in standard form having center at (1, 2), one focus at (6, 2) and passing through the point (4, 6).<\/p>\n Solution : <\/span>With center at (1, 2), the equation of the ellipse is \\((x – 1)^2\\over a^2\\) + \\((y – 2)^2\\over b^2\\) = 1. It passes through the point (4, 6) Find the equation of ellipse whose foci are (2, 3), (-2, 3) and whose semi major axis is of length \\(\\sqrt{5}\\).<\/a><\/p>\n![\"ellipse\"](\"https:\/\/i0.wp.com\/mathemerize.com\/wp-content\/uploads\/2021\/07\/ellipse.png?resize=590%2C350&ssl=1\")
<\/p>\n(a) Equation of directrix of Ellipse<\/strong> :\u00a0<\/h4>\n
(b) Major axis of Ellipse<\/strong> :\u00a0<\/h4>\n
(c) Minor axis of Ellipse<\/strong> :<\/h4>\n
(d)\u00a0 Double ordinate of Ellipse<\/strong> :<\/h4>\n
(e)\u00a0 Latus Rectum of Ellipse<\/strong> :<\/h4>\n
\n
L1′(-ae, -\\(b^2\\over a\\))<\/p>\n<\/blockquote>\n(f)\u00a0 Eccentricity of Ellipse<\/strong> :<\/h4>\n
\n\\(\\implies\\) \\(9\\over a^2\\) + \\(16\\over b^2\\) = 1 …..(i)
\nDistance between focus and center = (6 – 1) = 5 = ae
\n\\(\\implies\\) \\(b^2\\) = \\(a^2\\) – \\(a^2e^2\\) = \\(a^2\\) – 25 …..(ii)
\nSolving (i) and (ii)
\nwe get \\(a^2\\) = 45 and \\(b^2\\) = 20
\nHence, the equation of the ellipse is \\((x – 1)^2\\over 45\\) + \\((y – 2)^2\\over 20\\) = 1<\/p>\n\n\n
\nRelated Questions<\/h3>\n