Mathematics

Ellipse Session - 1

Session Objectives

1.I ntroduction 2.S tandard form of ellipse 3.D efinition of special points or lines 4.D efinition in form of focal length 5.P arametric form, eccentric angle 6.P osition of point with respect to ellipse 7.I ntersection of line and ellipse 8.C ondition for tangency 9.E quation of tangent in slope form, point of contact

Ellipse (i)Fixed point is known as focus (S). (ii)Fixed line is known as directrix (DD´). (iii) Fixed ratio is known as eccentricty (e). 0 < e < 1 It is the locus of a point P(h, k) in x-y plane which moves such that the ratio of its distance from a fixed point to its distance from a fixed straight line is constant.

Equation of Ellipse in Standard Form

Equation of Ellipse in Second Form As we have already discussed that in the equation, then the major and minor axes lie along x-axis and y-axis respectively. But if, then the major axis of the ellipse lies along y-axis and is of length 2b and minor axis along the x-axis and is of length 2a.

Definition of Special Points/Lines of the Ellipse Ellipse (I) Coordinates of the centre(0, 0) (II) Coordinates of the vertices (III) Coordinates of foci (IV) Length of major axis 2a 2b

Ellipse (V) Equation of the directrices (VII) Equation of minor axisx = 0y = 0 (VI) Equation of major axis y = 0 x = 0 Definition of Special Points/Lines of the Ellipse

Ellipse (VI) Eccentricity (VII) Length of the latus rectum Definition of Special Points/Lines of the Ellipse

Focal Distance of a Point on the Ellipse Let P(x, y) be any point on the ellipse Then SP = ePN

Focal Distance of a Point on the Ellipse SP = a – ex... (i) and S´P = ePN´ = e(RZ´) = e(OR + OZ´) = Major axis “An ellipse is the locus of a point which moves in such a way that the sum of its distances from two fixed points (foci) is always constant.” On the basis of above property, the definition of ellipse can be given as follows.

General Equation of Ellipse If the centre of the ellipse is at point (h, k) and the axes of ellipse is parallel to the coordinate axes, then its equation is.

Parametric Form of Ellipse Auxiliary circle The circle described on the major axis of an ellipse as diameter is called an auxiliary circle of the ellipse. If is an ellipse, then its auxiliary circle is x 2 + y 2 = a 2.

Eccentric Angle of Point Q lies on the circle, coordinate of

Parametric Coordinates of a Point on an Ellipse Let coordinates of P be. lies on ellipse

Equation of Chord

Position of a point w.r.t. ellipse E (0, 0) = –1 = –ve E (0, 1) = i.e. (i) If point (x 1, y 1 ) lies inside the ellipse, then E(x 1, y 1 ) < 0. (ii) If point (x 1, y 1 ) lies on the ellipse, then E(x 1, y 1 ) = 0. (iii) If point (x 1, y 1 ) lies outside the ellipse, then E(x 1, y 1 ) > 0.

Intersection of Line and Ellipse Let line mx – y + c = 0 and ellipse intersect at the distinct points A.

Intersection of Line and Ellipse

If discriminant of that quadratic > 0, then the line intersect the ellipse at two distinct points. If discriminant of that quadratic = 0, the line touches the ellipse. If discriminant of that quadratic < 0, the line does not cut the ellipse.

Intersection of Line and Ellipse Now, let us consider the case when D = 0.

Intersection of Line and Ellipse is always tangent to the ellipse for all values of

Point of contact of a tangent with Let us consider again the equation. Let the point of contact be, i.e. or

Point of contact of a tangent with Also y = mx + c is tangent line, passing through point of contact. where

Point of contact of a tangent with so that (if m is positive) (if m is positive) and the tangent touches the ellipse at point where so that (if ‘m’ is positive), (if ‘m’ is positive). Also we note that these two tangents are parallel.

Class Exercise - 1 Find the centre, vertices, lengths of axes, eccentricity, coordinates of foci, equations of directrices, and length of latus-rectum of the ellipse

Solution We have Shifting the origin at (3, 1) without rotating the coordinate axes, i.e. put X = x – 3 and Y = y – 1

Solution contd.. Equation (i) reduces to Clearly, a > b. Therefore, the given equation represents an ellipse whose major and minor axes are along X-axis and Y-axis respectively. Centre: The coordinates of the centre with respect to new axes are X = 0 and Y = 0. Coordinates of the centre with respect to old axes are x – 3 = 0 and y – 1 = 0, i.e. (3, 1).

Solution contd.. Vertices: The coordinates of vertices with respect to the new axes are The vertices with respect to the old axes are given by Lengths of axes: Here a = 4, b = 2 Length of major axis = 2a = 8 Length of minor axis = 2b = 4

Solution contd.. Eccentricity: The eccentricity e is given by Coordinates of foci: The coordinates of foci with respect to new axes are Coordinates of foci with respect to old axes are

Solution contd.. Equation of directrices: The equation of directrices with respect to new axes are, Equation of directrices with respect to old axes are

Class Exercise - 2 Find the equation of the ellipse whose axes are parallel to the coordinate axes respectively having its centre at the point (2, –3), one focus at (3, –3) and one vertex at (4, –3).

Solution Let 2a and 2b be the major and minor axes of the ellipse. Then its equation is As we know that distance between centre and vertex is the semi-major axes, Again, since the distance between the focus and centre is equal to ae,

Solution contd.. Again Equation of ellipse is

Class Exercise - 3 An ellipse has OB as a semi minor axis. F and F´ are its foci and is a right angle. Find the eccentricity of ellipse.

Solution The equation of the ellipse is Coordinates of F and F´ are (ae, 0) and (–ae, 0) respectively. Coordinates of B are (0, b).

Solution contd.. Slope of BF = and slope of BF´ = is right angle,

Class Exercise - 4 Let P be a variable point on the ellipse with foci at S and S´. If A be the area of PSS´, find the maximum value of A.

Solution Here equation of ellipse is Coordinates of P can be taken as

Solution contd.. Coordinates of Maximum area = 12 sq. unit as maximum value of

Class Exercise - 5 Find the equation of tangents to the ellipse which cut off equal intercepts on the axes.

Solution In case of tangent makes equal intercept makes equal intercepts on the axes, then it is inclined at an angle of to X-axis and hence its slope is Equation of tangent is

Class Exercise - 6 Find the equation of tangent to the ellipse which are (i) parallel, (ii) perpendicular to the line y + 2x = 4.

Solution Equation of ellipse can be written Slope of the line y = –2x + 5 is –2. Any tangent to the ellipse is If the tangent is parallel to the given line, slope of tangent is –2.

Solution contd.. Equation of tangent is If the tangent is perpendicular to the given line, slope of tangent is. Equation of tangent is

Class Exercise - 7 Prove that eccentric angles of the extremities of latus recta of the ellipse are given by

Solution Let be the eccentric angle of an end of a latus rectum. Then the coordinates of the end of latus rectum is. As we know that coordinates of latus rectum is,

Class Exercise - 8 A circle of radius r is concentric with the ellipse Prove that the common tangent is inclined to the major axis at an angle.

Solution Equation to the circle of radius r and concentric with ellipse whose centre is (0, 0) is Any tangent to the ellipse is If it is a tangent to circle, then perpendicular from centre (0, 0) is equal to r.

Solution contd..

Class Exercise - 9 If the line lx + my + n = 0 will cut the ellipse in points whose eccentric angles differ by then prove that

Solution Let the line lx + my + n = 0 cuts the ellipse at These two points lie on the line lx + my + n = 0 and Squaring and adding,

Class Exercise - 10 Find the locus of the foot of the perpendicular drawn from centre upon any tangent to the ellipse

Solution Any tangent to the given ellipse is Equation of any line perpendicular to (i), passing through the origin is

Solution contd.. In order to find the locus of P, the point of intersection of (i) and (ii), we have to eliminate m. This is the required locus.

Thank you