Current Affairs JEE Main & Advanced

Let the equation of ellipse in standard form will be given by \[\frac{{{x}^{2}}}{{{a}^{2}}}+\frac{{{y}^{2}}}{{{b}^{2}}}=1\].    Then the equation of ellipse in the parametric form will be given by \[x=a\cos \varphi ,y=b\sin \varphi \], where \[\varphi \] is the eccentric angle whose value vary from \[0\le \varphi <2\pi \]. Therefore coordinate of any point P on the ellipse will be given by \[(a\cos \varphi \,,\,b\sin \varphi )\].

(1) If the centre of the ellipse is at point \[(h,k)\] and the directions of the axes are parallel to the coordinate axes, then its equation is \[\frac{{{(x-h)}^{2}}}{{{a}^{2}}}+\frac{{{(y-k)}^{2}}}{{{b}^{2}}}=1\].     (2) If the equation of the curve is \[\frac{{{(lx+my+n)}^{2}}}{{{a}^{2}}}\] \[+\frac{{{(mx-ly+p)}^{2}}}{{{b}^{2}}}=1\], where \[lx+my+n=0\] and \[mx-ly+p=0\] are perpendicular lines, then we substitute \[\frac{lx+my+n}{\sqrt{{{l}^{2}}+{{m}^{2}}}}=X,\] \[\frac{mx-ly+p}{\sqrt{{{l}^{2}}+{{m}^{2}}}}=Y\], to put the equation in the standard form.

Let \[P({{x}_{1}},{{y}_{1}})\]be any point and let \[\frac{{{x}^{2}}}{{{a}^{2}}}+\frac{{{y}^{2}}}{{{b}^{2}}}=1\] is the equation of an ellipse. The point lies outside, on or inside the ellipse as if \[{{S}_{1}}=\frac{x_{1}^{2}}{{{a}^{2}}}+\frac{y_{1}^{2}}{{{b}^{2}}}-1>,\,\,=,<0\]    

The line \[y=mx+c\] intersects the ellipse \[\frac{{{x}^{2}}}{{{a}^{2}}}+\frac{{{y}^{2}}}{{{b}^{2}}}=1\] in two distinct points if \[{{a}^{2}}{{m}^{2}}+{{b}^{2}}>{{c}^{2}}\], in one point if \[{{c}^{2}}={{a}^{2}}{{m}^{2}}+{{b}^{2}}\] and does not intersect if \[{{a}^{2}}{{m}^{2}}+{{b}^{2}}<{{c}^{2}}\].  

(1) Point form: The equation of the tangent to the ellipse \[\frac{{{x}^{2}}}{{{a}^{2}}}+\frac{{{y}^{2}}}{{{b}^{2}}}=1\] at the point \[({{x}_{1}},{{y}_{1}})\] is \[\frac{x{{x}_{1}}}{{{a}^{2}}}+\frac{y{{y}_{1}}}{{{b}^{2}}}=1\].     (2) Slope form: If the line \[y=mx+c\]touches the ellipse \[\frac{{{x}^{2}}}{{{a}^{2}}}+\frac{{{y}^{2}}}{{{b}^{2}}}=1\], then \[{{c}^{2}}={{a}^{2}}{{m}^{2}}+{{b}^{2}}\]. Hence, the straight line \[y=mx\pm \sqrt{{{a}^{2}}{{m}^{2}}+{{b}^{2}}}\]always represents the tangents to the ellipse.     Points of contact: Line \[y=mx\pm \sqrt{{{a}^{2}}{{m}^{2}}+{{b}^{2}}}\] touches the ellipse \[\frac{{{x}^{2}}}{{{a}^{2}}}+\frac{{{y}^{2}}}{{{b}^{2}}}=1\] at  \[\left( \frac{\pm {{a}^{2}}m}{\sqrt{{{a}^{2}}{{m}^{2}}+{{b}^{2}}}},\frac{\mp {{b}^{2}}}{\sqrt{{{a}^{2}}{{m}^{2}}+{{b}^{2}}}} \right)\].       (3) Parametric form: The equation of tangent at any point \[(a\cos \varphi ,b\sin \varphi )\] is \[\frac{x}{a}\cos \varphi +\frac{y}{b}\sin \varphi =1\].

Pair of tangents: The equation of pair of tangents PA and PB is \[S{{S}_{1}}={{T}^{2}}\],         where \[S\equiv \frac{{{x}^{2}}}{{{a}^{2}}}+\frac{{{y}^{2}}}{{{b}^{2}}}-1\]   \[{{S}_{1}}\equiv \frac{x_{1}^{2}}{{{a}^{2}}}+\frac{y_{1}^{2}}{{{b}^{2}}}-1\]   \[T\equiv \frac{x{{x}_{1}}}{{{a}^{2}}}+\frac{y{{y}_{1}}}{{{b}^{2}}}-1\]   Director circle: The director circle is the locus of points from which perpendicular tangents are drawn to the ellipse.   Hence locus of \[P({{x}_{1}},{{y}_{1}})\]i.e., equation of director circle is \[{{x}^{2}}+{{y}^{2}}={{a}^{2}}+{{b}^{2}}\].

(1) Point form: The equation of the normal at \[({{x}_{1}},{{y}_{1}})\]to the ellipse \[\frac{{{x}^{2}}}{{{a}^{2}}}+\frac{{{y}^{2}}}{{{b}^{2}}}=1\]is \[\frac{{{a}^{2}}x}{{{x}_{1}}}-\frac{{{b}^{2}}y}{{{y}_{1}}}={{a}^{2}}-{{b}^{2}}\].     (2) Parametric form: The equation of the normal to the ellipse \[\frac{{{x}^{2}}}{{{a}^{2}}}+\frac{{{y}^{2}}}{{{b}^{2}}}=1\] at \[(a\cos \varphi ,b\sin \varphi )\] is \[ax\sec \varphi -by\,\text{cos}\text{ec}\varphi =\] \[{{a}^{2}}-{{b}^{2}}\].     (3) Slope form: If m is the slope of the normal to the ellipse \[\frac{{{x}^{2}}}{{{a}^{2}}}+\frac{{{y}^{2}}}{{{b}^{2}}}=1\], then the equation of normal is \[y=mx\pm \frac{m({{a}^{2}}-{{b}^{2}})}{\sqrt{{{a}^{2}}+{{b}^{2}}{{m}^{2}}}}\].     The co-ordinates of the point of contact are  \[\left( \frac{\pm {{a}^{2}}}{\sqrt{{{a}^{2}}+{{b}^{2}}{{m}^{2}}}},\frac{\pm m{{b}^{2}}}{\sqrt{{{a}^{2}}+{{b}^{2}}{{m}^{2}}}} \right)\] .

The circle described on the major axis of an ellipse as diameter is called an auxiliary circle of the ellipse.     If \[\frac{{{x}^{2}}}{{{a}^{2}}}+\frac{{{y}^{2}}}{{{b}^{2}}}=1\] is an ellipse, then its auxiliary circle is \[{{x}^{2}}+{{y}^{2}}={{a}^{2}}\].     Eccentric angle of a point: Let P be any point on the ellipse \[\frac{{{x}^{2}}}{{{a}^{2}}}+\frac{{{y}^{2}}}{{{b}^{2}}}=1\]. Draw PM perpendicular from P on the major axis of the ellipse and produce MP to meet the auxiliary circle in Q. Join CQ. The angle \[\angle XCQ=\varphi \] is called the eccentric angle of the point P on the ellipse.     Note that the angle \[\angle XCP\] is not the eccentric angle of point P.