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Mathematics

Mathematics. Ellipse Session - 1. Session Objectives. Session Objectives. I ntroduction S tandard form of ellipse D efinition of special points or lines D efinition in form of focal length P arametric form, eccentric angle P osition of point with respect to ellipse

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Mathematics

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  1. Mathematics

  2. Ellipse Session - 1

  3. Session Objectives

  4. Session Objectives • Introduction • Standard form of ellipse • Definition of special points or lines • Definition in form of focal length • Parametric form, eccentric angle • Position of point with respect to ellipse • Intersection of line and ellipse • Condition for tangency • Equation of tangent in slope form, point ofcontact

  5. 0 < e < 1 Ellipse It is the locus of a point P(h, k) inx-y plane which moves such that theratioof its distance from a fixed pointto itsdistance from a fixed straightline isconstant. • Fixed point is known as focus (S). • Fixed line is known as directrix (DD´). • Fixed ratio is known as eccentricty (e).

  6. Equation of Ellipse in Standard Form

  7. As we have already discussed that inthe equation, thenthe 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. Equation of Ellipse in Second Form

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

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

  10. Ellipse Ellipse (VI) Eccentricity (VII) Length of the latusrectum Definition of Special Points/Lines of the Ellipse

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

  12. = Major axis Focal Distance of a Point on the Ellipse SP = a – ex ... (i)and S´P = ePN´ = e(RZ´) = e(OR + OZ´) On the basis of above property, the definition of ellipse can be given as follows. “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.”

  13. If the centre of the ellipse is at point(h, k) and the axes of ellipse is parallelto the coordinate axes, then its equationis . General Equation of Ellipse

  14. If is an ellipse, then its auxiliary circle is x2 + y2 = a2. Parametric Form of Ellipse Auxiliary circle The circle described on themajoraxis of an ellipse asdiameter iscalled an auxiliarycircle of theellipse.

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

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

  17. Equation of Chord

  18. Equation of Chord

  19. Equation of Chord

  20. E (0, 1) = Position of a point w.r.t. ellipse E (0, 0) = –1 = –ve i.e. (i) If point (x1, y1) lies inside the ellipse, then E(x1, y1) < 0. (ii)If point (x1, y1) lies on the ellipse, then E(x1, y1) = 0. (iii)If point (x1, y1) lies outside the ellipse, then E(x1, y1) > 0.

  21. Let line mx – y + c = 0 and ellipse intersectat the distinct pointsA. Intersection of Line and Ellipse

  22. Intersection of Line and Ellipse

  23. Intersection of Line and Ellipse If discriminant of that quadratic > 0,thenthe line intersect the ellipseat two distinctpoints. If discriminant of that quadratic = 0,the line touches the ellipse. If discriminant of that quadratic < 0,the line does not cut the ellipse.

  24. Intersection of Line and Ellipse Now, let us consider the case whenD = 0.

  25. is always tangent to the ellipsefor all values of Intersection of Line and Ellipse

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

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

  28. 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 tangentsare parallel. Point of contact of a tangent with

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

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

  31. Equation (i) reduces to Coordinates of the centre with respect to old axes are x – 3 = 0 and y – 1 = 0, i.e. (3, 1). Solution contd.. 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.

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

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

  34. Equation of directrices: The equationof directriceswith respect to new axesare , Equation of directrices with respect to oldaxesare Solution contd..

  35. 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).

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

  37. Again Equation of ellipse is Solution contd..

  38. 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. Class Exercise - 3

  39. 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

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

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

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

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

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

  45. In case of tangent makes equal interceptmakes equal intercepts on the axes, thenit is inclined at an angle of to X-axisandhence its slope is Equation of tangent is Solution

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

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

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

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

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

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