Conic Section FINAL 05.01

1 VLet the relation of knowledge to real life be very visible to your pupilsand let them understand how by knowledge the world could betransformed. BERTRAND RUSSELL the preceding Chapter 10, we have studied various formsof the equations of a line. In this Chapter, we shall studyabout some other curves, viz., circles, ellipses, parabolasand hyperbolas. The names parabola and hyperbola aregiven by Apollonius. These curves are in fact, known asconic sections or more commonly conics because theycan be obtained as intersections of a plane with a doublenapped right circular cone. These curves have a very widerange of applications in fields such as planetary motion,design of telescopes and antennas, reflectors in flashlightsand automobile headlights, etc.

2 Now, in the subsequent sections we will see how theintersection of a plane with a double napped right circular coneresults in different types of of a ConeLet l be a fixed vertical line and m be another line intersecting it ata fixed point V and inclined to it at an angle ( ).Suppose we rotate the line m around the line l in such a waythat the angle remains constant. Then the surface generated isa double-napped right circular hollow cone herein after referred asApollonius(262 -190 )11 ChapterFig 11. 1 Conic SECTIONS2022-23 Conic SECTIONS 237 Fig 11. 2 Fig 11. 3cone and extending indefinitely far in both directions ( ).The point V is called the vertex; the line l is the axis of the cone.

3 The rotating linem is called a generator of the cone. The vertex separates the cone into two partscalled we take the intersection of a plane with a cone, the Section so obtained is calleda Conic Section . 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 theintersecting plane with respect to the cone and by the angle made by it with the verticalaxis of the cone. Let be the angle made by the intersecting plane with the verticalaxis of the cone ( ).The intersection of the plane with the cone can take place either at the vertex ofthe cone or at any other part of the nappe either below or above the Circle, ellipse, parabola and hyperbola When the plane cuts the nappe (otherthan the vertex) of the cone, we have the following situations:(a)When = 90o, the Section is a circle ( ).

4 (b)When < < 90o, the Section is an ellipse ( ).(c)When = ; the Section is a parabola ( ).(In each of the above three situations, the plane cuts entirely across one nappe ofthe cone).(d)When 0 < ; the plane cuts through both the nappes and the curves ofintersection is a hyperbola ( ).2022-23238 MATHEMATICSFig 11. Degenerated Conic sectionsWhen the plane cuts at the vertex of the cone, we have the following different cases:(a)When < 90o, then the Section is a point ( ).(b)When = , the plane contains a generator of the cone and the Section is astraight line ( ).It is the degenerated case of a parabola.

5 (c)When 0 < , the Section is a pair of intersecting straight lines ( ). It isthe degenerated case of a 11. 6 Fig 11. 7 Fig 11. 52022-23 Conic SECTIONS 239In the following sections, we shall obtain the equations of each of these conicsections in standard form by defining them based on geometric 11. 8 Fig 11. 9 Fig 11. 1 A circle is the set of all points in a plane that are equidistant from a fixedpoint in the fixed point is called the centre of the circle and the distance from the centreto a point on the circle is called the radius of the circle (Fig ).2022-23240 MATHEMATICSThe equation of the circle is simplest if the centre of the circle is at the , we derive below the equation of the circle with a given centre and radius(Fig ).

6 Given C (h, k) be the centre and r the radius of circle. Let P(x, y) be any point onthe circle ( ). Then, by the definition, | CP | = r . By the distance formula,we have22()()x hy kr+= (x h)2 + (y k)2 = r2 This is the required equation of the circle with centre at (h,k) and radius r .Example 1 Find an equation of the circle with centre at (0,0) and radius Here h = k = 0. Therefore, the equation of the circle is x2 + y2 = 2 Find the equation of the circle with centre ( 3, 2) and radius Here h = 3, k = 2 and r = 4. Therefore, the equation of the required circle is(x + 3)2 + (y 2)2 = 16 Example 3 Find the centre and the radius of the circle x2 + y2 + 8x + 10y 8 = 0 Solution The given equation is(x2 + 8x) + (y2 + 10y) = 8 Now, completing the squares within the parenthesis, we get(x2 + 8x + 16) + (y2 + 10y + 25) = 8 + 16 + (x + 4)2 + (y + 5)2 = {x ( 4)}2 + {y ( 5)}2 = 72 Therefore, the given circle has centre at ( 4, 5) and radius 11.

7 11 Fig 11. 122022-23 Conic SECTIONS 241 Example 4 Find the equation of the circle which passes through the points (2, 2), and(3,4) and whose centre lies on the line x + y = Let the equation of the circle be (x h)2 + (y k)2 = the circle passes through (2, 2) and (3,4), we have(2 h)2 + ( 2 k)2 = (1)and(3 h)2 + (4 k)2 = (2)Also since the centre lies on the line x + y = 2, we haveh + k = (3)Solving the equations (1), (2) and (3), we geth = , k = and r2 = , the equation of the required circle is(x )2 + (y )2 = each of the following Exercises 1 to 5, find the equation of the circle (0,2) and radius ( 2,3) and radius (41,21) and radius (1,1) and radius ( a, b) and radius 22ba.

8 In each of the following Exercises 6 to 9, find the centre and radius of the (x + 5)2 + (y 3)2 = + y2 4x 8y 45 = + y2 8x + 10y 12 = + 2y2 x = the equation of the circle passing through the points (4,1) and (6,5) andwhose centre is on the line 4x + y = the equation of the circle passing through the points (2,3) and ( 1,1) andwhose centre is on the line x 3y 11 = the equation of the circle with radius 5 whose centre lies on x-axis andpasses through the point (2,3). the equation of the circle passing through (0,0) and making intercepts a andb on the coordinate the equation of a circle with centre (2,2) and passes through the point (4,5).

9 The point ( , ) lie inside, outside or on the circle x2 + y2 = 25?2022-23242 MATHEMATICSFig 11. 13 Fig ParabolaDefinition 2 A parabola is the set of all pointsin a plane that are equidistant from a fixed lineand a fixed point (not on the line) in the fixed line is called the directrix ofthe parabola and the fixed point F is called thefocus (Fig ). ( Para means for and bola means throwing , , the shapedescribed when you throw a ball in the air).ANote If the fixed point lies on the fixedline, then the set of points in the plane, whichare equidistant from the fixed point and thefixed line is the straight line through the fixedpoint and perpendicular to the fixed line.

10 Wecall this straight line as degenerate case ofthe line through the focus and perpendicularto the directrix is called the axis of theparabola. The point of intersection of parabolawith the axis is called the vertex of the parabola( ). St andard equations of parabola Theequation of a parabola is simplest if the vertexis at the origin and the axis of symmetry is along the x-axis or y-axis. The four possiblesuch orientations of parabola are shown below in (a) to (d).2022-23 Conic SECTIONS 243We will derive the equation for the parabola shown above in Fig (a) withfocus at (a, 0) a > 0; and directricx x = a as below:Let F be the focus and l the directrix.