CONIC SECTIONS

1 SECTIONS of a cone Let l be a fixed vertical line and m be another line intersectingit at a fixed point V and inclined to it at an angle (Fig. ).Fig. we rotate the line m around the line l in such a way that the angle remainsconstant. Then the surface generated is a double-napped right circular hollow coneherein after referred as cone and extending indefinitely in both directions (Fig. ).Fig. SECTIONS18/04/18 The point V is called the vertex; the line l is the axis of the cone. The rotating line m iscalleda generator of the cone. The vertex separates the cone into two parts we take the intersection of a plane with a cone, the section so obtained is called aconic section .

2 Thus, CONIC SECTIONS are the curves obtained by intersecting a rightcircular cone by a obtain different kinds of CONIC SECTIONS depending on the position of the intersectingplane with respect to the cone and the angle made by it with the vertical axis of thecone. Let be the angle made by the intersecting plane with the vertical axis of thecone ( ).The intersection of the plane with the cone can take place either at the vertex of thecone or at any other part of the nappe either below or above the the plane cuts the nappe (other than the vertex) of the cone, we have thefollowing situations:(a)When = 90o, the section is a circle.(b)When < < 90o, the section is an ellipse.

3 (c)When = ; the section is a parabola.(In each of the above three situations, the plane cuts entirely across one nappeof the cone).(d)When 0 < ; the plane cuts through both the nappes and the curves ofintersection is a these curves are important tools for present day exploration of outer space andalso for research into the behaviour of atomic take CONIC SECTIONS as plane curves. For this purpose, it is convenient to use equivalentdefinition that refer only to the plane in which the curve lies, and refer to special pointsand lines in this plane called foci and directrices. According to this approach, parabola,ellipse and hyperbola are defined in terms of a fixed point (called focus) and fixed line(called directrix) in the S is the focus and l is the directrix, then the set of all points in the plane whosedistance from S bears a constant ratio e called eccentricity to their distance from l is aconic special case of ellipse, we obtain circle for which e = 0 and hence we study Circle A circle is the set of all points in a plane which are at a fixed distancefrom a fixed point in the plane.

4 The fixed point is called the centre of the circle and thedistance from centre to any point on the circle is called the radius of the SECTIONS 18718/04/18188 EXEMPLAR PROBLEMS MATHEMATICSFig. equation of a circle with radius r havingcentre (h, k) is given by (x h)2 + (y k)2 = r2 The general equation of the circle is given byx2 + y2 + 2gx + 2fy + c = 0, where g, f and c areconstants.(a)The centre of this circle is ( g, f)(b)The radius of the circle is 22gfc+ The general equation of the circle passing throughthe origin is given by x2 + y2 + 2gx + 2fy = equation of second degree , ax2 + 2hxy + by2 + 2gx + 2fy + c = 0 representa circle if (i) the coefficient of x2 equals the coefficient of y2, , a = b 0 and (ii) thecoefficient of xy is zero, , h = parametric equations of the circle x2 + y2 = r2 are given by x = r cos , y = r sin where is the parameter and the parametric equations of the circle (x h)2 + (y k)2 = r2are given byx h =r cos , y k = r sin orx =h + r cos , y = k + r sin.

5 Fig. :The general equation of the circle involves threeconstants which implies that at least three conditions arerequired to determine a circle ParabolaA parabola is the set of points P whose distances from afixed point F in the plane are equal to their distances froma fixed line l in the plane. The fixed point F is called focusand the fixed line l the directrix of the SECTIONS 189 Standard equations of parabolaThe four possible forms of parabola are shown below in Fig. (a) to (d)The latus rectum of a parabola is a line segment perpendicular to the axis of theparabola, through the focus and whose end points lie on the parabola (Fig.)

6 Fig. facts about the parabolaForms of Parabolasy2 = 4axy2 = 4axx2 = 4ayx2 = 4ayAxisy = 0y = 0x = 0x = 0 Directixx = ax = ay = ay = aVertex(0, 0)(0, 0)(0, 0)(0, 0)Focus(a, 0)( a, 0)(0, a)(0, a)Length of latus4a4a4a4arectumEquations of latusx = ax = ay = ay = arectum18/04/18190 EXEMPLAR PROBLEMS MATHEMATICSF ocal distance of a pointLet the equation of the parabola be y2 = 4ax and P(x, y) be a point on it. Then thedistance of P from the focus (a, 0) is called the focal distance of the point, ,FP =22()x ay +=2()4x aax +=2()x a+=| x + a | Ellipse An ellipse is the set of points in a plane, the sum of whose distancesfrom two fixed points is constant.

7 Alternatively, an ellipse is the set of all points in theplane whose distances from a fixed point in the plane bears a constant ratio, less than,to their distance from a fixed line in the plane. The fixed point is called focus, the fixedline a directrix and the constant ratio (e) the centricity of the have two standard forms of the ellipse, ,(i)22221xyab+=and(ii)22221xyba+=,In both cases a > b and b2 = a2(1 e2), e < (i) major axis is along x-axis and minor along y-axis and in (ii) major axis is along y-axis and minor along x-axis as shown in Fig. (a) and (b) facts about the EllipseFig. SECTIONS 191 Forms of the ellipse22221xyab+=22221xyba+=a > ba > bEquation of major axisy = 0x = 0 Length of major axis2a 2aEquation of Minor axisx = 0y = 0 Length of Minor axis2b2bDirectricesx = aey = aeEquation of latus rectumx = aey = aeLength of latus rectum22ba22baCentre(0, 0)(0, 0)Focal DistanceThe focal distance of a point (x, y)

8 On the ellipse 22221xyab+= isa e | x | from the nearer focusa + e | x | from the farther focusSum of the focal distances of any point on an ellipse is constant and equal to the lengthof the major Hyperbola A hyperbola is the set of all points in a plane, the difference ofwhose distances from two fixed points is constant. Alternatively, a hyperbola is the setof all points in a plane whose distances from a fixed point in the plane bears a constantratio, greater than 1, to their distances from a fixed line in the plane. The fixed point iscalled a focus, the fixed line a directrix and the constant ratio denoted by e, the ecentricityof the have two standard forms of the hyperbola, ,(i)22221xyab = and(ii)22221yxab =18/04/18192 EXEMPLAR PROBLEMS MATHEMATICSHere b2 = a2 (e2 1), e > (i) transverse axis is along x-axis and conjugate axis along y-axis where as in (ii)transverse axis is along y-axis and conjugate axis along facts about the HyperbolaForms of the hyperbola22221xyab =22221yxab =Equation of transverse axisy = 0x = 0 Equation of conjugate axisx = 0y = 0 Length of transverse axis2a2aFoci( ae, 0)(0, ae)

9 Equation of latus rectumx = aey = aeLength of latus rectum22ba22baCentre(0, 0)(0, 0)18/04/18 CONIC SECTIONS 193 Focal distanceThe focal distance of any point (x, y) on the hyperbola 22221xyab = ise | x | a from the nearer focuse | x | + a from the farther focusDifferences of the focal distances of any point on a hyperbola is constant and equal tothe length of the transverse equation of conicsConicsParametric equations(i)Parabola : y2 = 4axx = at2, y = 2at; < t < (ii)Ellipse : 22221xyab+=x = a cos , y = b sin ; 0 2 (iii)Hyperbola : 22221xyab =x = a sec , y = b tan , where;22 < < 322 < < Solved ExamplesShort Answer T ypeExample 1 Find the centre and radius of the circle x2 + y2 2x + 4y = 8 Solution we write the given equation in the form (x2 2x) + (y2 + 4y) = 8 Now, completing the squares, we get(x2 2x + 1) + (y2 + 4y + 4) = 8 + 1 + 4(x 1)2 + (y + 2)2 = 13 Comparing it with the standard form of the equation of the circle, we see that thecentre of the circle is (1, 2)

10 And radius is 2 If the equation of the parabola is x2 = 8y, find coordinates of the focus,the equation of the directrix and length of latus The given equation is of the form x2 = 4ay where a is , the focus is on y-axis in the negative direction and parabola opens EXEMPLAR PROBLEMS MATHEMATICSC omparing the given equation with standard form, we get a = , the coordinates of the focus are (0, 2) and the the equation of directrix isy = 2 and the length of the latus rectum is 4a, , 3 Given the ellipse with equation 9x2 + 25y2 = 225, find the major and minoraxes, eccentricity, foci and We put the equation in standard form by dividing by 225 and get22259xy+ =1 This shows that a = 5 and b = 3.