4 Session Objectives Equation of the tangent in point (x1, y1) form Equation of normal in point (x1, y1) formEquation of tangent and normal in parametric formNumber of tangents drawn from a point to an ellipseDirector circleEquation of pair of tangentsEquation of chord of contactEquation of normal in slope (m) formNumber of normals (co-normal points)Equation of chord whose middle point is givenDiameter of ellipseConjugate diameters
5 Equation of the Tangent in Point (x1, y1) Form Equation of tangent to ellipse at (x1, y1) is given by
6 Equation of the Tangent in Point (x1, y1) Form Working rule for finding T = 0,replace and keeping constant unchanged.
7 Equation of Normal in Point (x1, y1) Form Equation of normal at to ellipse given by
8 Equation of Tangent and Normal in Parametric () Form Equation of tangent at using point form isSlope point of contact with the ellipse.
9 Equation of Tangent and Normal in Parametric () Form Equation of normal at becomesSlope = foot of normal
10 Number of Tangents Drawn From a Point to an Ellipse Two tangents can be drawn from a point to an ellipse it may be real or imaginary.
11 Number of Tangents Drawn From a Point to an Ellipse Real and distinct if , i.e. point lies outside the ellipse.Real and coincident if , i.e. point lies on the ellipse.Imaginary if , i.e. point lies inside the ellipse.
12 Director CircleDirector circle is the locus of point of intersection of perpendicular tangents to the conic.Equation of director circle of ellipseLet (h, k) be the point of intersection of tangents to and slope of tangent be m. Then we have
13 Director Circle if tangents are perpendicular, then , i.e. Hence, locus of (h, k) is
14 Equation of Pair of Tangents As we have seen earlier from a point (h, k) lying outside the ellipse, we have two real and distinct tangents possible. Combined equation of these tangents is given byor have usual meanings.Note: We can obtain above equation by eliminating ‘m’ fromy – k = m(x – h) and
15 Equation of Chord of Contact Equation of chord of contact of point (h, k) outside the ellipse isor T = 0PQ is chord of contact of point (h, k).
16 Equation of Normal Slope (m) Form Equation of normal of slope m to isSlope = m, foot of normal isNumber of Normals (Co-normal Points)Four normals can be drawn from a point to an ellipse.
17 Equation of Chord Whose Middle Point is Given Equation of chord of bisected at (h, k) is , i.ewhere T, S1 have usual meanings.
18 Diameter of EllipseThe locus of mid-point of a system of parallel chords of an ellipse is called diameter and chords are called its double ordinates. The end points of the diameter lying on the ellipse are called vertices of diameter.Equation of diameter of ellipseLet the system of parallel chords be given by y = mx + c, where ‘m’ is fixed and ‘c’ is a variable. Let (h, k) be its middle point. Then equation of chord with middle point at (h, k) is given by
19 Diameter of Ellipse Then its slope is m. Hence, Locus of (h, k) is or which is the required equation of diameter. Note that diameter passes through the centre of ellipse. Hence, equation of diameter bisecting the parallel chords of slope ‘m’ of ellipse
20 Conjugate diametersTwo diameters of an ellipse are said to be conjugate diameters. If each bisects the chords parallel to the other.Condition of conjugate diametersLet be two conjugate diameters of (Recall that diameter of ellipse passes through the centre of the ellipse.). Then diameter bisecting the chords parallel to is given bywhich is given as y= m2x.
21 Conjugate diameters Then or Thus, are conjugate diameters of ellipse . Note: Major axis and minor axis are conjugate diameters, as each bisects the chords parallel to the other but product of their slopes is not defined.
22 Properties of conjugate diameters The eccentric angles of the ends of a pair of conjugate diameters of an ellipse differ by
23 Properties of conjugate diameters (ii)The sum of the squares of any two conjugate semi-diameters (half of the diameter) is constant and is given by sum of squares of semi-axis, i.e.Note: That major axis and minor axis are also conjugate diameters.(iii)The product of the focal distances of a point on an ellipse is equal to the square of the semi-diameter which is conjugate to the diameter passing through this point.
24 Properties of conjugate diameters (iv) Tangents at the ends of the pair of conjugate diameters form a parallelogram, i.e. ABCD is a parallelogram.
25 Properties of conjugate diameters (v) The area of the parallelogram formed by the tangents at the ends of conjugate diameters is constant and is given by the product of the axes, i.e. area (ABCD) = 4ab.
26 Concyclic PointsAny circle intersects an ellipse in two or four real points. They are called concyclic points.If a, b, g, d be the eccentric angles of four concyclic points on an ellipse, thena + b + g + d = 2np, i.e. even multiple of p.
27 Class Exercise - 1The area of the quadrilateral formed by the tangents at the end points of latus rectum to the ellipse is(a) sq. units (b) 9 sq. units (c) 27 sq. units (d) sq. units
28 SolutionEnd points of latus rectum are given by
29 Solution contd..Tangents at these points, using point form, are given byEquation of AD = 2x + 3y = 9Area of the parallelogram ABCD= 4 × AreaHence, answer is (c).
30 Class Exercise - 2Find the point on which is nearest to the line x + y = 7.
31 SolutionThe point which is nearest to the line is the point in the first quadrant, where tangent is parallel to PQ or if PQ is moved parallel to itself towards ellipse, the point where PQ first meets, i.e. touches the ellipse is the required point.
32 Solution contd.. Let be the required point, tangent at is Also lies on (2, 1) is the required point.Note: (–2, –1) is the farthest from the line x + y = 7
33 Solution contd.. Alternative: Slope of tangent is –1. Point of contact isRequired point is (2, 1).
34 Class Exercise - 3If the normal at the end of a latus rectum of an ellipse of eccentricity ‘e’ passes through one end of the minor axis, then(a) 1 (b) 2 (c) 3 (d) 4
47 Class Exercise - 7Find the locus of the middle points of chord of an ellipse which are drawn through the positive end of the minor axis.
48 SolutionLet (a cos , b sin ) be the coordinate of the other extremities of the chord of ellipse The positive end of the minor axis is clearly (0, b).Let (x, y) be the mid-point of the chord.
49 Solution contd.. Squaring and adding (i) and (ii), On simplifying the required locus of (x, y) is
50 Class Exercise - 8Prove that the area of the triangle form by three points on an ellipse, whose eccentric angles are isProve also that its area is to the area of the triangle formed by the corresponding points on the auxiliary circle as b : a.
51 Solution Let the coordinates of the given points on the ellipse be be Area of the triangle formed by these points
52 Solution contd.. Using Second Part: Clearly, coordinates of the points of the auxiliary circle corresponding to the given three points may be obtained by putting b = a.
53 Class Exercise - 9Prove that the straight line is a normal to the ellipse if
54 Solution The normal at any point of the ellipse is given by Comparing the given line, and the equation of the normal