In mensuration we shall study about some other curves, viz., circles, ellipses, parabolas and hyperbolas. The names parabola and hyperbola are given by Apollonius. These curves are in fact, known as conic sections or more commonly conics because they can be obtained as intersections of a plane with a double napped right circular cone. These curves have a very wide range of applications in fields such as planetary motion, design of telescopes and antennas, reflectors in flashlights and automobile headlights, etc. Now, in the subsequent sections we will see how the intersection of a plane with a double napped right circular cone results in different types of curves.

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). 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 far in both directions (Fig.11.2)

The point V is called the vertex; the line l is the axis of the cone. The rotating line m is called a 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 obtained 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 intersecting plane with respect to the cone and by the angle made by it with the vertical axis of the cone. Let β be the angle made by the intersecting plane with the vertical axis of the cone (Fig11.3). 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.

Introduction to Conic Sections

 

Circle, Ellipse, Parabola and Hyperbola

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 (Fig 11.4)
(b) When α < β < 90o , the section is an ellipse (Fig 11.5)
(c) When β = α; the section is a parabola (Fig 11.6)
(In each of the above three situations, the plane cuts entirely across one nappe of the cone).
(d) When 0 ≤ β < α; the plane cuts through both the nappes and the curves of intersection is a hyperbola (Fig11.7)

Conic Sections: Intro to Circles

Conic Sections: Intro to Ellipses

Conic Sections: Intro to Hyperbolas

 

Degenerated conic sections

When the plane cuts at the vertex of the cone, we have the following different cases:

(a) When α < β ≤ 90o , then the section is a point (Fig 11.8)
(b) When β = α, the plane contains a generator of the cone and the section is a straight line (Fig 11.9)
It is the degenerated case of a parabola.
(c) When 0 ≤ β < α, the section is a pair of intersecting straight lines (Fig 11.10) It is the degenerated case of a hyperbola.

In the following sections, we shall obtain the equations of each of these conic sections in standard form by defining them based on geometric properties.

Circle 1: A circle is the set of all points in a plane that are equidistant from a fixed point in the plane.

The fixed point is called the centre of the circle and the distance from the centre to a point on the circle is called the radius of the circle (Fig 11.11)

The equation of the circle is simplest if the centre of the circle is at the origin. However, we derive below the equation of the circle with a given centre and radius (Fig 11.12)

Given C (h, k) be the centre and r the radius of circle. Let P(x, y) be any point on the circle .Then, by the definition, | CP | = r . By the distance formula, we have

$\sqrt{\left ( x-h \right )^{2}+\left ( y-k \right )^{2}}=\: r$

${\left ( x-h \right )^{2}+\left ( y-k \right )^{2}}=\: r^{2}$

This is the required equation of the circle with centre at (h,k) and radius r .

Example: Find the equation of the circle with centre (–3, 2) and radius 4.

Solution: Here h = –3, k = 2 and r = 4. Therefore, the equation of the required circle is

${\left ( x+3 \right )^{2}+\left ( y-2 \right )^{2}}=\: 4^{2}$

= ${\left ( x+3 \right )^{2}+\left ( y-2 \right )^{2}}=\: 16$

Example 2: Find the centre and the radius of the circle $x^{2}+y^{2}+8x+10y-8=\: 0$

Solution: The given equation is$(x^{2}+8x)+y^{2}+10y=\: 8$

Now, completing the squares within the parenthesis, we get

=$\left ( 8^{2} +8x+16\right )+\left ( y^{2}+10y+25 \right )=\: 8+16+25$$

=$\left ( x+4 \right )^{2}+\left ( y+5 \right )^{2}=\: 49$

=$\left \{ x-(-4)^{2} \right \}+\left \{ y-(-5)^{2} \right \}=\: 7^{2}$

therefore, the given circle has centre at (– 4, –5) and radius 7

 

Parabola

2 A parabola is the set of all points in a plane that are equidistant from a fixed line and a fixed point (not on the line) in the plane. The fixed line is called the directrix of the parabola and the fixed point F is called the focus (Fig 11.13) ‘Para’ means ‘for’ and ‘bola’ means ‘throwing’, i.e., the shape described when you throw a ball in the air).

Note: If the fixed point lies on the fixed line, then the set of points in the plane, which are equidistant from the fixed point and the fixed line is the straight line through the fixed point and perpendicular to the fixed line. We call this straight line as degenerate case of the parabola.

A line through the focus and perpendicular to the directrix is called the axis of the parabola. The point of intersection of parabola with the axis is called the vertex of the parabola (Fig 11.14)

Standard Equations of Parabola: The equation of a parabola is simplest if the vertex is at the origin and the axis of symmetry is along the x-axis or y-axis. The four possible such orientations of parabola are shown below in (Fig 11.15)

where PB is perpendicular to l. The coordinates of B are (– a, y). By the distance formula, we have

$PF=\: \sqrt{(x-a)^{2}+y^{2}}$ and $PB=\: \sqrt{(x+a)^{2}}$

since PF = PB we get ,

$\sqrt{(x-a)^{2}+y^{2}}\:= \:sqrt{(x+a)^{2}}$

${(x-a)^{2}+y^{2}}\:=\: {(x+a)^{2}}$ or $x^{2}-2ax+a^{2}+b^{2}\:=\: x^{2}+2ax+a^{2}$

or $y^{2}\: =\: 4ax$ (a > 0)

Hence, any point on the parabola satisfies or $y^{2}\: =\: 4ax$

Note: The standard equations of parabolas have focus on one of the coordinate axis; vertex at the origin and thereby the directrix is parallel to the other coordinate axis. However, the study of the equations of parabolas with focus at any point and any line as directrix is beyond the scope here.

From the standard equations of the parabolas, we have the following observations:

1. Parabola is symmetric with respect to the axis of the parabola.If the equation has a $y^{2}$ term, then the axis of symmetry is along the x-axis and if the equation has an $x^{2}$term, then the axis of symmetry is along the y-axis.

2. When the axis of symmetry is along the x-axis the parabola opens to the
(a) right if the coefficient of x is positive,
(b) left if the coefficient of x is negative.

3. When the axis of symmetry is along the y-axis the parabola opens
(c) upwards if the coefficient of y is positive.
(d) downwards if the coefficient of y is negative.

 

Latus Rectum

Latus rectum of a parabola is a line segment perpendicular to the axis of the parabola, through the focus and whose end points lie on the parabola (Fig 11.17)

To find the Length of the latus rectum of the parabola $y^{2}$= 4ax (Fig 11.18)

By the definition of the parabola, AF = AC.

But AC = FM = 2a

Hence AF = 2a.

And since the parabola is symmetric with respect to x-axis AF = FB and so AB = Length of the latus rectum = 4a.

Example: Find the coordinates of the focus, axis, the equation of the directrix and latus rectum of

$y^{2\:}=\: 8x$
Solution: The given equation involves $y^{2}$, so the axis of symmetry is along the x-axis.

The coefficient of x is positive so the parabola opens to the right. Comparing with the given equation $y^{2} \:= \:4ax$ , we find that a = 2.

Thus, the focus of the parabola is (2, 0) and the equation of the directrix of the parabola is x = – 2 (Fig 11.19)

Length of the latus rectum is 4a = 4 × 2 = 8.

 

Ellipse

An ellipse is the set of all points in a plane, the sum of whose distances from two fixed points in the plane is a constant. The two fixed points are called the foci (plural of ‘focus’) of the ellipse (Fig 11.10)

Note : The constant which is the sum of the distances of a point on the ellipse from the two fixed points is always greater than the distance between the two fixed points.

The mid point of the line segment joining the foci is called the centre of the ellipse. The line segment through the foci of the ellipse is called the major axis and the line segment through the centre and perpendicular to the major axis is called the minor axis. The end points of the major axis are called the vertices of the ellipse.(Fig 11.21)

We denote the length of the major axis by 2a, the length of the minor axis by 2b and the distance between the foci by 2c. Thus, the length of the semi major axis is a and semi-minor axis is b

 

Relationship between semi-major axis, semi-minor axis and the distance of the focus from the centre of the ellipse

Relationship between semi-major axis, semi-minor axis and the distance of the focus from the centre of the ellipse (Fig 11.23). [/su_heading]

Take a point P at one end of the major axis. Sum of the distances of the point P to the foci is

F1 P + F2 P = F1 O + OP + F2 P (Since, F1 P = F1 O + OP) = c + a + a – c = 2a

Take a point Q at one end of the minor axis. Sum of the distances from the point Q to the foci is $2F_{1}Q\:\:F_{2}Q=\sqrt{a^{2}+c^{2}}+\sqrt{b^{2}+c^{2}}=\sqrt[2]{b^{2}+a^{2}}$

Since both P and Q lies on the ellipse. By the definition of ellipse, we get,

$\sqrt[2]{b^{2}+c^{2}}=2a$i.e$\sqrt[a]{b^{2}+c^{2}}$

$a^{2}=b^{2}+c^{2}$i.e$a\sqrt{a^{2}-b^{2}}$

Special Cases of an Ellipse In the equation $c^{2}={\left ( a^{2}-b^{2} \right )}$ obtained above, if we keep a fixed and vary c from 0 to a, the resulting ellipses will vary in shape.

Case (i) When c = 0, both foci merge together with the centre of the ellipse and $a^{2}\: =\: b^{2 }$ , i.e., a = b, and so the ellipse becomes circle (Fig 11.24 )Thus, circle is a special case of an ellipse .

Case (ii) When c = a, then b = 0. The ellipse reduces to the line segment F1F2 joining the two foci (Fig 11.25)

Eccentricity The eccentricity of an ellipse is the ratio of the distances from the centre of the ellipse to one of the foci and to one of the vertices of the ellipse (eccentricity is denoted by e i.e $e\: =\: \frac{c}{a}$

Then since the focus is at a distance of c from the centre, in terms of the eccentricity the focus is at a distance of ae from the centre.

Standard Equations of an Ellipse: The equation of an ellipse is simplest if the centre of the ellipse is at the origin and the foci are on the x-axis or y-axis. The two such possible orientations are shown in Fig 11.26

Note: From the equation of the ellipse obtained above, it follows that for every point P (x, y) on the ellipse, we get$\frac{x^{2}}{a^{2}}=1-\frac{y^{2}}{b^{2}}\: i.e\:x^{2}\leq a^{2},so\: -a\leq x\leq a$

Therefore, the ellipse lies between the lines x = – a and x = a and touches these lines. Similarly, the ellipse lies between the lines y = – b and y = b and touches these lines

we can also derive the equation of the ellipse as $\frac{x^{2}}{b^{2}}+\frac{y^{2}}{a^{2}}\: =\: 1$

These two equations are known as standard equations of the ellipses.

Note : The standard equations of ellipses have centre at the origin and the major and minor axis are coordinate axes. However, the study of the ellipses with centre at any other point, and any line through the centre as major and the minor axes passing through the centre and perpendicular to major axis are beyond the scope here.

From the Standard Equations of the Ellipses we have the following Observations:

1. Ellipse is symmetric with respect to both the coordinate axes since if (x, y) is a point on the ellipse, then (– x, y), (x, –y) and (– x, –y) are also points on the ellipse.

2. The foci always lie on the major axis. The major axis can be determined by finding the intercepts on the axes of symmetry. That is, major axis is along the x-axis if the coefficient of $x^{2}$ has the larger denominator and it is along the y-axis if the coefficient of $x^{2}$ has the larger denominator.

Latus Rectum: Latus rectum of an ellipse is a line segment perpendicular to the major axis through any of the foci and whose end points lie on the ellipse Fig 11.28.

To Find the Length of the Latus Rectum of the Ellipse: $\frac{x^{2}}{a^{2}}+\frac{y^{2}}{b^{2}}=\: \: 1$

Let the length of $AF_{2}$ be l. Then the coordinates of A are (c, l ),i.e.,(ae,l)

Since the ellipse is symmetric with respect to y-axis (of course, it is symmetric w.r.t. both the coordinate axes), $AF_{2}\: =\: F_{2}B$and so length of the latus rectum is$\frac{2b^{2}}{a}$

Example: Find the coordinates of the foci, the vertices, the length of major axis, the minor axis, the eccentricity and the latus rectum of the ellipse . $\frac{x^{2}}{25}+\frac{y^{2}}{9}\: =\: 1$

Since denominator of $\frac{x^{2}}{25}$ is larger than the denominator of $\frac{y^{2}}{9}$,the major axis is along the x-axis. Comparing the given equation with $\frac{x^{2}}{25}+\frac{y^{2}}{9}\: =\: 1$ , we get a = 5 and b = 3. Also

$c=\sqrt{a^{2}-b^{2}}=\sqrt{5^{2}-9^{2}}=\sqrt{25-9}$ = 4

 

Hyperbola

Hyperbola: A hyperbola is the set of all points in a plane, the difference of whose distances from two fixed points in the plane is a constant. The term “difference” that is used in the definition means the distance to the farther point minus the distance to the closer point. The two fixed points are Called the Foci of the Hyperbola.

The mid-point of the line segment joining the foci is called the centre of the hyperbola. The line through the foci is called the transverse axis and the line through the centre and perpendicular to the transverse axis is called the conjugate axis. The points at which the hyperbola intersects the transverse axis are called the vertices of the hyperbola (Fig 11.29).

We denote the distance between the two foci by 2c, the distance between two vertices (the length of the transv

erse axis) by 2a and we define the quantity b as

$b=\sqrt{c^{2}- b^{2}}$ also 2b is the length of the conjugate axis (Fig 11.30)

Eccentricity: Just like an ellipse, the ratio e = c/a is called the eccentricity of the hyperbola. Since c ≥ a, the eccentricity is never less than one. In terms of the eccentricity, the foci are at a distance of ae from the centre.

Standard Equation of Hyperbola: The equation of a hyperbola is simplest if the centre of the hyperbola is at the origin and the foci are on the x-axis or y-axis. The two such possible orientations are shown in Fig 11.31

The Equation of Hyperbola with Origin (0,0) and Transverse Axis :

$\frac{x^{2}}{a^{2}}-\frac{y^{2}}{b{^{2}}}\: =1\:$

These two equations are known as the standard equations of hyperbolas.

Note: A hyperbola in which a = b is called an equilateral hyperbola.

The standard equations of hyperbolas have transverse and conjugate axes as the coordinate axes and the centre at the origin. However, there are hyperbolas with any two perpendicular lines as transverse and conjugate axes, but the study of such cases will be dealt in higher classes.

From the standard equation of hyperbola ,we have find observations :

1. Hyperbola is symmetric with respect to both the axes, since if (x, y) is a point on the hyperbola, then (– x, y), (x, – y) and (– x, – y) are also points on the hyperbola.
2. The foci are always on the transverse axis. It is the positive term whose denominator gives the transverse axis.

For example, $\frac{x^{2}}{3^{2}}-\frac{y^{2}}{4{^{2}}}\: =1\:$ has transverse axis along x-axis of length 6, while $\frac{x^{2}}{5^{2}}-\frac{y^{2}}{2{^{2}}}\: =1\:$ has transverse axis along y-axis of length 10.

 

Mensuration Questions from Previous Year Exam