Let S and S^' denote the foci, SL the semi latus rectum of...

Let $S$ and $S^{\prime}$ denote the foci, SL the semi latus rectum of an ellipse. LS' produced cuts the ellipse at $P$. Show that the length of the ordinate of $\mathrm{P}$ is $\frac{\left(1-\mathrm{e}^{2}\right)^{2}}{\left(1+3 \mathrm{e}^{2}\right)}$ a where, $2 \mathrm{a}$ is the length of the major axis and $e$ is the eccentricity of the ellipse.

Key Concepts

Coordinate Geometry in Conic Sections

Coordinate geometry provides a framework for describing conic sections using equations, which allows for the precise calculation of properties like axes lengths, foci locations, and ordinates. By translating geometric figures into algebraic forms, many problems concerning ellipses and other conic sections can be analyzed and solved using systematic methods.

Semi-Latus Rectum

The semi-latus rectum is a segment associated with conic sections, particularly significant in ellipses. It is the distance from a focus to the curve measured along a line perpendicular to the major axis. This parameter is used in formulas relating to the ellipse’s curvature and in deriving relationships involving other measurements, such as ordinates.

Ellipse

An ellipse is a conic section defined as the set of all points in a plane for which the sum of the distances to two fixed points (the foci) is constant. This shape is characterized by its major and minor axes, and its unique geometric properties form the basis for many problems in analytic geometry, physics, and engineering.

Foci

The foci are the two fixed points used in the definition of an ellipse. Their locations determine the ellipse’s eccentricity and overall shape. In many geometric constructions and proofs involving ellipses, the foci play a crucial role in linking the algebraic and geometric properties of the conic.

Eccentricity

Eccentricity is a measure of how much an ellipse deviates from being a circle. It is given by the ratio of the distance from the center to a focus over the semi-major axis length. The value of the eccentricity, which ranges between 0 and 1 for ellipses, directly influences the ellipse’s shape and is key in deriving various other geometric properties.

Transcript

00:01 In this problem of circle and conic section, we have given that let s and s prime, so this is s and s prime, denote the foci and this is, so this is the foci and sl is the semi -latus rectum, so which is sl is semi -litus rectum, lattice -rectum of an ellipse.

00:27 So here we have given the conic is ellipse.

00:31 L s prime so this is l s prime produced cuts the ellipse at point p so this is cuts at point p point p we have to show that the length of the ordinate p is say 1 minus e square whole square divided with 1 plus 3 e square where this is multiplied with a where 2a is the length of major axis and e is the eccentricity so here 2a is the length of major axis axis and here e is eccentricity of the conic section or we can say ellipse so here first we have to consider that the equation so here let the ellipse be ellipse b a is the major axis so we can say x squared divided with a square plus y squared divided with b square is equal to 1 semi -lector rectum so this would be b squared divided with a is equal to a multiplied with 1 minus e e square coordinate of lr so here we can say coordinate of l r so this is a e and a multiplied with 1 minus e e square and we know that s is we can say s is ae 0 and we can say s prime is this would be minus ae so this is minus ae and 0.

02:28 Now slope of l s so this is slope of ls prime so we can say slope of ls prime is equals to this is a multiplied with 1 minus e squared divided with twice of a e or we can cut this term and we can say this is 1 minus e squared divided with 2 e now equation so equation of ls prime would be y is equal to say this is y is equal to this term is 1 minus e squared divided with 2 e and this would be x plus a e now we can say this is our equation number 1.

03:16 So this is your equation number 1 and from equation number 1 so we can say from equation number 1 we can say this is x is equals to twice of e y divided with 1 minus e square and this would be minus of a so minus ae and now we have to substitute this in equation of ellipse so so this is the value of x.

03:45 So this would be equation of ellipse as this is twice of ey divided with 1 minus e square minus ae and this would be whole square and this is multiplied with 1 divided with a squared plus y divided with b is equal to 1.

04:09 So this is your equation number 2...

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