Ellipse

Importance

An ellipse is a curve on a plane surrounding two focal points such that a straight line drawn from one of the focal points to any point on the curve and then back to the other focal point has the same length for every point on the curve.

Yearly trend

Generally, the questions are asked on tangents and normals of the standard form of ellipse. Observing the number of questions asked in JEE-Mains and JEE-Advanced in previous years, one can easily expect a qustion or two in further years as well.

Concepts

The standard equation of an ellipse is given by !!(x^2/a^2)+(y^2/b^2)=1!!,
where 2a and 2b are lengths of the major and minor axis with eccentricity(e)!!=sqrt(1-(b^2/a^2))!!. The latus rectum of a standard ellipse is given by !!x=+-ae!!.

Equation of tangent: The equation of tangent of the standard ellipse, !!x^2/a^2+y^2/b^2=1!! at the point !!(x_1,y_1)!! is given by !!(x x_1)/a^2+(y y_1)/b^2=1!!

Equation of normal: The equation of normal of the standard ellipse, !!x^2/a^2+y^2/b^2=1!! is at the point !!(x_1,y_1)!! is given by !!(x a^2)/(x_1)-(y b^2)/y_1=a^2e^2!!

Also, the parametric coordinates of a point on the ellipse is a vary important concept, i.e. !!(acostheta,bsintheta)!!.

As it is in case of all conics, the properties of tangents and normals are really crucial as they can help reduce a lot of effort in solving a problem.