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11.1 Overview

11.1.1 Sections of a cone Let l be a fixed vertical line and m be another line intersecting

it at a fixed point V and inclined to it at an angle α (Fig. 11.1).Fig. 11.1 Suppose we rotate the line m around the line l in such a way that the angle α remains constant. Then the surface generated is a double-napped right circular hollow cone herein after referred as cone and extending indefinitely in both directi ons (Fig. 11.2).Fig. 11.2Fig. 11.3Chapter 11

CONIC SECTIONS

The point V is called the vertex; the line l is the axis of the cone. The rotating line m is calleda generator of the cone. The vertex separates the cone into two parts called nappes. If we take the intersection of a plane with a cone, the section so obtai ned is called a conic section. Thus, conic sections are the curves obtained by intersecting a right circular cone by a plane. We obtain different kinds of conic sections depending on the position of the intersec ting plane with respect to the cone and the angle made by it with the vertica l axis of the cone. Let β be the angle made by the intersecting plane with the vertical axis of t he cone (Fig.11.3). The intersection of the plane with the cone can take place either at the vertex of the cone or at any other part of the nappe either below or above the vertex. When the plane cuts the nappe (other than the vertex) of the cone, we have the following situations: (a)When β = 90o, the section is a circle. (b)When α <

β < 90o, the section is an ellipse.

(c)When β = α; the section is a parabola. (In each of the above three situations, the plane cuts entirely across one nappe of the cone). intersection is a hyperbola. Indeed these curves are important tools for present day exploration of o uter space and also for research into the behaviour of atomic particles. We take conic sections as plane curves. For this purpose, it is convenien t to use equivalent definition that refer only to the plane in which the curve lies, and ref er to special points and 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 foc us) and fixed line (called directrix) in the plane. If S is the focus and l is the directrix, then the set of all points in the plane whose distance from S bears a constant ratio e called eccentricity to their distance from l is a conic section. As special case of ellipse, we obtain circle for which e = 0 and hence we study it differently.

11.1.2 Circle A circle is the set of all points in a plane which are at a fixed distanc

e from a fixed point in the plane. The fixed point is called the centre of the circle and the distance from centre to any point on the circle is called the radius of the circle.CONIC SECTIONS 187

188 EXEMPLAR PROBLEMS - MATHEMATICSFig. 11.4The equation of a circle with radius r having

centre ( h, k) is given by (x - h)2 + (y - k)2 = r2

The general equation of the circle is given by

x

2 + y2 + 2gx + 2fy + c = 0, where g, f and c are

constants. (a)The centre of this circle is (- g, - f)

(b)The radius of the circle is 2 2g f c+ -The general equation of the circle passing throughthe origin is given by x2 + y2 + 2gx + 2fy = 0.

General equation of second degree i.e., ax2 + 2hxy + by2 + 2gx + 2 fy + c = 0 represent

a circle if (i) the coefficient of x2 equals the coefficient of y2, i.e., a = b ≠ 0 and (ii) the

coefficient of xy is zero, i.e., h = 0. The parametric equations of the circle x2 + y2 = r2 are given by x = r cosθ, y = r sinθ where θ is the parameter and the parametric equations of the circle (x - h)2 + (y - k)2 = r2 are given by x - h =r cosθ, y - k = r sinθ orx =h + r cosθ, y = k + r sinθ.Fig. 11.5 Note:The general equation of the circle involves three constants which implies that at least three conditions are required to determine a circle uniquely.

11.1.3 Parabola

A parabola is the set of points P whose distances from a fixed point F in the plane are equal to their distances from a fixed line l in the plane. The fixed point F is called focus and the fixed line l the directrix of the parabola.Fig. 11.6 CONIC SECTIONS 189Standard equations of parabola The four possible forms of parabola are shown below in Fig. 11.7 (a) to (d) The latus rectum of a parabola is a line segment perpendicular to the ax is of the parabola, through the focus and whose end points lie on the parabola (F ig. 11.7).

Fig. 11.7

Main facts about the parabolaForms of Parabolasy2 = 4axy2 = - 4axx2 = 4ayx2 = - 4ayAxisy = 0y = 0x = 0x = 0Directixx = - ax = ay = - a y = aVertex(0, 0) (0, 0)(0, 0)(0, 0)Focus(a, 0)(- a, 0)(0, a)(0, - a)

Length of latus4a4a4a4arectumEquations of latusx = ax = - a y = ay = - a rectum

190 EXEMPLAR PROBLEMS - MATHEMATICSFocal distance of a point

Let the equation of the parabola be y2 = 4ax and P(x, y) be a point on it. Then the distance of P from the focus (a, 0) is called the focal distance of the point, i.e.,

FP =2 2( )x a y- +=

2( ) 4x a ax- +=

2( )x a+=| x + a |

11.1.4 Ellipse An ellipse is the set of points in a plane, the sum of whose distances

from two fixed points is constant. Alternatively, an ellipse is the set of all points in the plane whose distances from a fixed point in the plane bears a constant r atio, less than, to their distance from a fixed line in the plane. The fixed point is cal led focus, the fixed line a directrix and the constant ratio (e) the centricity of the ellipse.

We have two standard forms of the ellipse, i.e.,

(i) 2 2 2 2 1x y a b+ =and(ii) 2 2 2 2 1x y b a+ =,

In both cases a > b and b2 = a2(1 - e2), e < 1.

In (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. 11.8 (a) and (b) respectively.

Main facts about the Ellipse

Fig. 11.8

CONIC SECTIONS 191Forms of the ellipse2 2

2 2 1x y a b+ = 2 2 2 2 1x y

b a+ =a > ba > bEquation of major axisy = 0x = 0Length of major axis2a 2aEquation of Minor axisx = 0y = 0Length of Minor axis2b2bDirectricesx = ±

a ey = ± a eEquation of latus rectumx = ± aey = ± aeLength of latus rectum 22b
a 22b
aCentre(0, 0)(0, 0)

Focal Distance

The focal distance of a point (x, y) on the ellipse 2 2 2 2 1x y a b+ = is a - e | x | from the nearer focus a + e | x | from the farther focus Sum of the focal distances of any point on an ellipse is constant and eq ual to the length of the major axis.

11.1.5 Hyperbola A hyperbola is the set of all points in a plane, the difference of

whose distances from two fixed points is constant. Alternatively, a hyperbola is the set of all points in a plane whose distances from a fixed point in the plane bears a constant ratio, greater than 1, to their distances from a fixed line in the plane . The fixed point is called a focus, the fixed line a directrix and the constant ratio denote d by e, the ecentricity of the hyperbola. We have two standard forms of the hyperbola, i.e., (i) 2 2 2 2 1x y a b- = and(ii) 2 2 2 2 1y x a b- =

192 EXEMPLAR PROBLEMS - MATHEMATICSHere b2 = a2 (e2 - 1), e > 1.

In (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 x-axis.

Fig. 11.9Main facts about the Hyperbola

Forms of the hyperbola2 2

2 2 1x y a b- = 2 2 2 2 1y x

a b- =Equation of transverse axisy = 0x = 0Equation of conjugate axisx = 0y = 0Length of transverse axis2a2aFoci(± ae, 0)(0, ± ae)Equation of latus rectumx = ± aey = ± aeLength of latus rectum

22b
a 22b
aCentre(0, 0)(0, 0)

CONIC SECTIONS 193

Focal distance

The focal distance of any point (x, y) on the hyperbola 2 2 2 2 1x y a b- = is e | x | - a from the nearer focus e | x | + a from the farther focus Differences of the focal distances of any point on a hyperbola is consta nt and equal to the length of the transverse axis.

Parametric equation of conics

ConicsParametric equations

(i)Parabola : y2 = 4axx = at2, y = 2at; - ∞ < t < ∞ (ii)Ellipse : 2 2 2 2 1x y (iii)Hyperbola : 2 2 2 2 1x y a b- =x = a secθ, y = b tanθ, where ;2 2

π π- < θ< 3

2 2

π π< θ<11.2 Solved Examples

Short Answer Type

Example 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 - 2 x + 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 se e that the centre of the circle is (1, -2) and radius is 13. Example 2 If the equation of the parabola is x2 = - 8y, find coordinates of the focus, the equation of the directrix and length of latus rectum.

Solution

The given equation is of the form x2 = - 4ay where a is positive. Therefore, the focus is on y-axis in the negative direction and parabola opens downwards.

194 EXEMPLAR PROBLEMS - MATHEMATICS

Comparing the given equation with standard form, we get a = 2. Therefore, the coordinates of the focus are (0, -2) and the the equ ation of directrix is y = 2 and the length of the latus rectum is 4a, i.e., 8.

Example 3

Given the ellipse with equation 9x2 + 25y2 = 225, find the major and minor axes, eccentricity, foci and vertices.

Solution

We put the equation in standard form by dividing by 225 and get 2 2 25 9
x y+ =1 This shows that a = 5 and b = 3. Hence 9 = 25(1 - e2), so e = 4

5. Since the denominator

of x2 is larger, the major axis is along x-axis, minor axis along y-axis, foci are (4, 0) and (- 4, 0) and vertices are (5, 0) and (-5, 0). Example 4 Find the equation of the ellipse with foci at (± 5, 0) and x = 36
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