ELLIPSE | QUICK REVISION | JEE Main | JEE Advanced | BITSAT – By Nitesh Choudhary - Videos

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In this video, we will revise the topic ELLIPSE.

I will start the topic by discussing the general equation of ellipse i.e. how to find the equation given its focus and equation of directrix. It is defined as the locus of a point which moves in such a way that the ratio of its distance from a fixed point and a fixed line is constant (eccentricity), which is less than one. Then, we will discuss standard equations of ellipse (referred to its principal axes along the co-ordinate axes) x^2/a^2 + y^2/b^2 = 1 and x^2/b^2 + y^2/a^2 = 1 (where a is greater than b) and will define the major axis and minor axis of the ellipse.

Next, we will discuss important parameters related to the equation of the ellipse – coordinates of focus, equation of directrix, focal chord, double ordinate, focal radii, latus rectum (shortest focal chord), Length of latus rectum, End points of latus rectum, Parametric form, Centre of ellipse, Area of Ellipse. How to find the coordinate of any point on the ellipse whose distance from focus i.e. focal radius is given? An important property of ellipse is that, sum of focal distances of any point P on the ellipse is constant and is equal to the length of major axis. We will also discuss about auxiliary circle (a circle described on major axis of the ellipse as diameter) and eccentric angle. Then, we will discuss topics related to a point and an ellipse. What are the conditions that needs to be applied if the point lies inside the ellipse, on the ellipse or outside the ellipse?

Next discussion will be related to equation of tangent to an ellipse. First topic is, what is the condition to be applied given a line is tangent to an ellipse or line is chord to an ellipse – slope form of tangent. How to find the equation of a tangent, point of contact on circle is given i.e. point form of tangent. We will also discuss equation of tangent in parametric form and the point of intersection of two tangents. The eccentric angles of the points of contact of two parallel tangents differ by π. Then, we will learn how to find the equation of tangents to an ellipse from an external point and angle between those tangents. Next is to find the locus of point of intersection of two perpendicular tangents i.e. Equation of Director Circle: x^2 + y^2 = a^2 + b^2. Then, we will discuss how to find the equation of pair of tangents to an ellipse. We will also some important results related to equation of tangents which are:
– The Locus Of Feet Of Perpendiculars From The Foci Upon Any Tangent Is Auxiliary Circle.
– The Product Of Perpendiculars From The Foci Upon Any Tangent Of The Ellipse Is b^2.
– The Length Of Tangent Between The Point Of Contact And The Point Where It Meets The Directrix Subtends Right Angle At The Corresponding Focus.
– The Tangents At The Extremities Of The Latus Rectum Of An Ellipse Intersect At The Foot Of The Corresponding Directrix And The Quadrilateral Thus Formed By Them Is A Rhombus Of Area: 2a^2/e.

Then, we will move onto discuss the equation of chord of an ellipse. In that, we will discuss the equation of chord and its slope in parametric form. Then, we will learn about some important properties related to focal chord – Semi-Latus Rectum is the harmonic mean of SP and SQ, where S is the focus and P, Q are the extremities of the focal chord. Then, we will discuss the equation of chord of contact and equation of chord with given middle point.

Next is the equation of normal to both the ellipses: x^2/a^2 + y^2/b^2 = 1 and x^2/b^2 + y^2/a^2 = 1 (where a is greater than b) – Slope form, Point Form and Parametric Form. We will discuss the condition that need to be applied such a given line is normal to a ellipse.

We will conclude the discussion with the important reflection property of an ellipse – Rays Emanating From One Focus After Reflecting On The Ellipse Passes Through Other Focus And Vice Versa Is Also True.

By – Nitesh Choudhary
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