{"id":3829,"date":"2021-08-09T08:19:55","date_gmt":"2021-08-09T08:19:55","guid":{"rendered":"https:\/\/mathemerize.com\/?p=3829"},"modified":"2022-02-15T22:49:47","modified_gmt":"2022-02-15T17:19:47","slug":"different-types-of-parabola","status":"publish","type":"post","link":"https:\/\/mathemerize.com\/different-types-of-parabola\/","title":{"rendered":"Different Types of Parabola Equations"},"content":{"rendered":"

Here, you will learn Different Types of Parabola and Standard equations of parabola, focal chord, double ordinate and latus rectum of parabola.<\/p>\n

Let’s begin –<\/p>\n

What is Parabola ?<\/h2>\n

A parabola is the locus of a point which moves in a plane, such that its distance from a fixed point(focus) is equal to its perpendicular distance from a fixed straight line(directrix).<\/p>\n

The Standard equation of parabola is \\(y^2 = 4ax\\) <\/strong>and it is shown in figure. For this parabola :<\/p>\n

(i) Vertex<\/strong> is (0,0).\"parabola\"<\/p>\n

(ii) focus<\/strong> is (a,0)<\/p>\n

(iii) Axis<\/strong> is y = 0<\/p>\n

(iv) Directrix<\/strong> is x + a = 0<\/p>\n

(a) Focal distance :<\/strong><\/p>\n

The distance of a point on the parabola from the focus is called the focal distance of the point.<\/p>\n

(b) Focal chord :<\/strong><\/p>\n

A chord of the parabola, which passes through the focus is called a focal chord.<\/p>\n

(c) Double ordinate :<\/strong><\/p>\n

A chord of the parabola perpendicular to the axis of the symmetry is called double ordinate.<\/p>\n

(d) Latus rectum :<\/strong><\/p>\n

A double ordinate passing through the focus or a focal chord perpendicular to the axis of parabola is called latus rectum.<\/p>\n

For \\(y^2 = 4ax\\).<\/p>\n

\n

Length of the latus rectum = 4a<\/p>\n

Length of the semi latus rectum = 2a<\/p>\n

Ends of the latus rectum are L(a, 2a) & L'(a, -2a).<\/p>\n<\/blockquote>\n

Note :<\/strong><\/p>\n

(i) Perpendicular distance from focus on the directrix = half the latus rectum.<\/em><\/p>\n

(ii) Vertex is middle point of the focus & point of intersection of directrix & axis.<\/em><\/p>\n

(iii) Two parabolas are said to be equal if they have the same latus rectum.<\/em><\/p>\n

Different Types of Parabola & Standard Equations of Parabola<\/h2>\n

Four different types of parabola equations are<\/p>\n

\n

\\(y^2\\) = 4ax ; \\(y^2\\) = -4ax ; \\(x^2\\) = 4ay ; \\(x^2\\) = -4ay.<\/p>\n<\/blockquote>\n

One I had shown above and three others are shown below.<\/p>\n\n\n

\n\t\t\t
\n\t\t\t \n\t\t\t

\\(y^2\\) = -4ax<\/b><\/p>\n\t\t\t <\/div>\n\t\t\t

\n\t\t\t \n\t\t\t

\\(x^2\\) = 4ay<\/b><\/p>\n\t\t\t <\/div>\n\t\t\t

\n\t\t\t \n\t\t\t

\\(x^2\\) = -4ay<\/b><\/p>\n\t\t\t <\/div>\n\t\t\t<\/div>
\n\n\n\n\n\t\t \n\t\t \n\t\t \n\t\t \n\t\t \n\t\t \n\t\t \n\t\t
Parabola<\/th>\n\t\t Vertex<\/th> \n\t\t Focus<\/th>\n\t\t Axis<\/th>\n\t\t Directrix<\/th>\n\t\t <\/tr>\n\t\t
\\(y^2\\) = 4ax<\/td>\n\t\t (0,0)<\/td> \n\t\t (a,0)<\/td>\n\t\t y = 0<\/td>\n\t\t x = -a<\/td>\n\t\t <\/tr>\n\t\t
\\(y^2\\) = -4ax<\/td>\n\t\t (0,0)<\/td> \n\t\t (-a,0)<\/td>\n\t\t y = 0<\/td>\n\t\t x = a<\/td>\n\t\t <\/tr>\n\t\t
\\(x^2\\) = +4ay<\/td>\n\t\t (0,0)<\/td> \n\t\t (0,a)<\/td>\n\t\t x = 0<\/td>\n\t\t y = -a<\/td>\n\t\t <\/tr>\n\t\t
\\(x^2\\) = -4ay<\/td>\n\t\t (0,0)<\/td> \n\t\t (0,-a)<\/td>\n\t\t x = 0<\/td>\n\t\t y = a<\/td>\n\t\t <\/tr>\n\t\t
\\((y-k)^2\\) = 4a(x-h)<\/td>\n\t\t (h,k)<\/td> \n\t\t (h+a,k)<\/td>\n\t\t y = k<\/td>\n\t\t x+a-h = 0<\/td>\n\t\t <\/tr>\n\t\t
\\((x-p)^2\\) = 4b(y-q)<\/td>\n\t\t (p,q)<\/td> \n\t\t (p,b+q)<\/td>\n\t\t x = p<\/td>\n\t\t y+b-q = 0<\/td>\n\t\t <\/tr>\n\t\t <\/tbody><\/table>
\n\n\t\t \n\t\t \n\t\t \n\t\t \n\t\t \n\t\t \n\t\t \n\t\t \n\t\t
Length of Latus rectum<\/th>\n\t\t Ends of Latus rectum<\/th>\n\t\t Parametric equation<\/th>\n\t\t Focal length<\/th>\n\t\t <\/tr>\n\t\t
4a<\/td>\n\t\t (a,\\(\\pm\\)2a)<\/td> \n\t\t (a\\(t^2\\), 2at)<\/td>\n\t\t x + a<\/td>\n\t\t <\/tr>\n\t\t
4a<\/td>\n\t\t (-a,\\(\\pm\\)2a)<\/td> \n\t\t (-a\\(t^2\\), 2at)<\/td>\n\t\t x – a<\/td>\n\t\t <\/tr>\n\t\t
4a<\/td>\n\t\t (\\(\\pm\\)2a,a)<\/td> \n\t\t (2at, a\\(t^2\\))<\/td>\n\t\t y + a<\/td>\n\t\t <\/tr>\n\t\t
4a<\/td>\n\t\t (\\(\\pm\\)2a,-a)<\/td> \n\t\t (2at, -a\\(t^2\\))<\/td>\n\t\t y – a<\/td>\n\t\t <\/tr>\n\t\t
4a<\/td>\n\t\t (h+a, k\\(\\pm\\)2a)<\/td> \n\t\t (h+a\\(t^2\\), k+2at)<\/td>\n\t\t x – h + a<\/td>\n\t\t <\/tr>\n\t\t
4b<\/td>\n\t\t (p\\(\\pm\\)2a, q+a)<\/td> \n\t\t (p+2at, q+a\\(t^2\\))<\/td>\n\t\t y – q + b<\/td>\n\t\t <\/tr>\n\t\t <\/tbody><\/table>\n\n\n\n

Example : <\/span> Find the vertex, axis, directrix, focus, latus rectum and the tangent at vertex for the parabola \\(9y^2 – 16x – 12y – 57\\) = 0.<\/p>\n

Solution : <\/span>The given equation can be written as \\(({y-2\\over 3})^2\\) = \n\t\t \\(16\\over 9\\)\\(({x + 61\\over 16})\\) which is of the form \\(y^2\\) = 4ax. Hence the vertex is (-\\(61\\over 16\\), \\(2\\over 3\\))

\n\t\t The axis is y – \\(2\\over 3\\) = 0 \\(\\implies\\) y = \\(2\\over 3\\)

\n\t\t The directrix is x + a – h = 0 \\(\\implies\\) x + \\(61\\over 16\\) + \\(4\\over 9\\) \\(\\implies\\) x = \\(-613\\over 144\\)

\n\t\t The focus is (h+a, k) \\(\\implies\\) (\\(-485\\over 144\\), \\(2\\over 3\\))

\n\t\t Length of the latus rectum = 4a = \\(16\\over 9\\)

\n\t\t The tangent at the vertex is x – h = 0 \\(\\implies\\) x = \\(-61\\over 16\\)

<\/p>\n\n\n

Position of a point relative to a parabola :<\/strong><\/p>\n

The point (\\(x_1\\),\\(y_1\\)) lies outside, on or inside the parabola \\(y^2\\) = 4a\\(x_1\\) is positive, zero or negative.<\/p>\n

\n

Related Questions<\/h3>\n

Find the value of k for which the point (k-1, k) lies inside the parabola \\(y^2\\) = 4x.<\/a><\/p>\n

The focal distance of a point on the parabola \\(y^2\\) = 12x is 4. Find the abscissa of this point.<\/a><\/p>\n

The slope of the line touching both the parabolas \\(y^2\\) = 4x and \\(x^2\\) = -32 is<\/a><\/p>\n\n\n

\n
Next – Equation of Tangent to Parabola in all Forms<\/a><\/div>\n<\/div>\n\n\n\n
\n
Previous – Graph of a Parabola \u2013 Types of Parabolas <\/a><\/div>\n<\/div>\n","protected":false},"excerpt":{"rendered":"

Here, you will learn Different Types of Parabola and Standard equations of parabola, focal chord, double ordinate and latus rectum of parabola. Let’s begin – What is Parabola ? A parabola is the locus of a point which moves in a plane, such that its distance from a fixed point(focus) is equal to its perpendicular …<\/p>\n

Different Types of Parabola Equations<\/span> Read More »<\/a><\/p>\n","protected":false},"author":1,"featured_media":0,"comment_status":"open","ping_status":"closed","sticky":false,"template":"","format":"standard","meta":{"site-sidebar-layout":"default","site-content-layout":"default","ast-global-header-display":"","ast-main-header-display":"","ast-hfb-above-header-display":"","ast-hfb-below-header-display":"","ast-hfb-mobile-header-display":"","site-post-title":"","ast-breadcrumbs-content":"","ast-featured-img":"","footer-sml-layout":"","theme-transparent-header-meta":"default","adv-header-id-meta":"","stick-header-meta":"","header-above-stick-meta":"","header-main-stick-meta":"","header-below-stick-meta":""},"categories":[12],"tags":[516,513,515,514,512],"yoast_head":"\nDifferent Types of Parabola Equations - Mathemerize<\/title>\n<meta name=\"description\" content=\"Learn Different Types of Parabola and Standard equations of parabola, focal chord, double ordinate and latus rectum of parabola here.\" \/>\n<meta name=\"robots\" content=\"index, follow, max-snippet:-1, max-image-preview:large, max-video-preview:-1\" \/>\n<link rel=\"canonical\" href=\"https:\/\/mathemerize.com\/different-types-of-parabola\/\" \/>\n<meta property=\"og:locale\" content=\"en_US\" \/>\n<meta property=\"og:type\" content=\"article\" \/>\n<meta property=\"og:title\" content=\"Different Types of Parabola Equations - 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