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Conic Sections

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Conic Sections

Definition:
The curves obtained by intersection of a plane and a double cone in different orientation are called conic section.
Conic Sections 1

Definitions of various important terms:

  1. Focus: The fixed point is called the focus of the conic-section.
    Conic Sections 2
  2. Directrix: The fixed straight line is called the directrix of the conic section.
  3. Eccentricity: The constant ratio is called the eccentricity of the conic section and is denoted by e.
    Conic Sections 3
  4. Axis: The straight line passing through the focus and perpendicular to the directrix is called the axis of the conic section. A conic is always symmetric about its axis.
  5. Vertex: The points of intersection of the conic section and the axis are called vertices of conic section.
  6. Centre: The point which bisects every chord of the conic passing through it, is called the centre of conic.
  7. Latus-rectum: The latus-rectum of a conic is the chord passing through the focus and perpendicular to the axis.
    Conic Sections 4
  8. Double ordinate: The double ordinate of a conic is a chord perpendicular to the axis.
  9. Focal chord: A chord passing through the focus of the conic is called a focal chord.
  10. Focal distance: The distance of any point on the conic from the focus is called the focal distance of the point.

General equation of a conic section when its focus, directrix and eccentricity are given:

Let S(α, β) be the focus, Ax + By + C = 0 be the directrix and e be the eccentricity of a conic.
Conic Sections 5Let P(h, k) be any point on the conic. Let PM be the perpendicular from P, on the directrix. Then by definition,
SP = ePM ⇒ SP2 = e2PM2
Conic Sections 6

Recognisation of conics

The equation of conics is represented by the general equation of second degree ax2 + 2hxy + by2 + 2gx + 2fy + c = 0 ……(i)
and discriminant of above equation is represented by ∆, where ∆ = abc + 2fgh – af2 – bg2 – ch2
Case I: When ∆ = 0.
In this case equation (i) represents the degenerate conic whose nature is given in the following table.

S. No.ConditionNature of conic
1.∆ = 0 and ab – h2 = 0A pair of coincident straight lines
2.∆ = 0 and ab – h2 < 0A pair of intersecting straight lines
3.∆ = 0 and ab – h2 > 0A point

Case II: When ∆ ≠ 0.
In this case equation (i) represents the non-degenerate conic whose nature is given in the following table.

S. No.ConditionNature of conic
1.∆ ≠ 0, h = 0, a = b, e = 0A circle
2.∆ ≠ 0, ab – h2 = 0, e = 1A parabola
3.∆ ≠ 0, ab – h2 > 0, e < 0An ellipse
4.∆ ≠ 0, ab – h2 < 0, e > 0A hyperbola
5.∆ ≠ 0, ab – h2 < 0, a + b = 0, e = √2A rectangular hyperbola

What are the Parts of a Circle

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What are the Parts of a Circle

So far, we have discussed about the triangle and quadrilateral that have linear boundaries. Circle is a closed figure that has a curvilinear boundary.
What are the Parts of a Circle 1When we think of circles, the very first thing that comes to our mind is its round shape, for example, bangles, coins, rings, plates, chapattis, pizzas, CDs etc. Wheels of a car, bus, cycle, truck, train, and aeroplane are also round in shape. If we take a stone, tie it to one end of a string and swing it in the air by holding the other end of the string, the path traced by the stone will be a circular path and it will make a circle.

Read More:

  1. Circle: A circle is a collection of all those points in a plane that are at a given constant distance from a given fixed point in the plane.
  2. Centre: Circle is a closed figure made up of points in a plane that are at the same distance from a fixed point, called the centre of the circle. In the figure O is the centre.
    What are the Parts of a Circle 3

 

  1. Radius: The constant distance from its centre is called the radius of the circle. In the figure, OA is radius
    What are the Parts of a Circle 2
  2. Chord: A line segment joining two points on a circle is called a chord of the circle. In the figure, AB is a chord of the circle. If a chord passes through centre then it is longest chord. PQ, PR, and ST are chords of the circle. Chord ST passes through the centre, hence it is a diameter.
    What are the Parts of a Circle 4
  3. Diameter: A chord passing through the centre of a circle is called the diameter of the circle. A circle has an infinite number of diameters. CD is the diameter of the circle as shown in the figure. If d is the diameter of the circle then d = 2r. where r is the radius. or the longest chord is called diameter.
    In the figure, AB is the diameter and the arcs CD and DC are semicircles.
    What are the Parts of a Circle 5
  4. Arc: A continuous piece of a circle is called an arc. Let A,B,C,D,E,F be the points on the circle. The circle is divided into different pieces. Then, the pieces AB, BC, CD, DE, EF etc. are all arcs of the circle.
    What are the Parts of a Circle 6Let P,Q be two points on the circle. These P, Q divide the circle into two parts. Each part is an arc. These arcs are denoted in anti-clockwise direction
    arc
  5. Circumference of a circle: The perimeter of a circle is called its circumference. The circumference of a circle of radius r is 2πr.
  6. Semicircle: The diameter of a circle divides the circle into two equal parts. Each part is called a semi-circle. We can also say that half of a circle is called a semi¬circle. In the figure,  AXB and AYB represents two semi-circles.
  7. Segment: Let AB be a chord of the circle. Then, AB divides the region enclosed by the circle (i.e., the circular disc) into two parts. Each of the parts is called a segment of the circle. The segment, containing the minor arc is called minor segment and the segment, containing the major arc, is called the major segment and segment of a circle is the region between an arc and chord of the circle.
    What are the Parts of a Circle 7
  8. Central Angles: Consider a circle. The angle subtended by an arc at the centre O is called the central angle. The vertex of the central angle is always at the centre O.
    What are the Parts of a Circle 8
  9. Degree measure of an arc: Degree measure of a minor arc is the measure of the central angle subtended by the arc.
    arc-1
    The degree measure of the circumference of the circle is always 360°.
  10. Interior and Exterior of Circle
    A circle divides the plane on which lies into three parts.
    (i) Inside the circle. which is called the interior of the circle
    (ii) Circle
    (iii) Outside the circle, which is called the exterior of the circle.
    The circle and its interior make up the circular region.
    What are the Parts of a Circle 9
  11. Sector:
    A sector is that region of a circular disc which lies between an arc and the two radii joining the extremities of the arc and the centre. OAB is a sector as shown in the figure.
    Quadrant: One fourth of a circular disc is called a quadrant.
    What are the Parts of a Circle 10
  12. Position of a point:
    Point Inside the circle: A point P, such that OP < r, is said to lie inside the circle.
    What are the Parts of a Circle 11The point inside the circle is also called interior point. (Example : Centre of cirle)
    Point outside the circle: A point Q, such that OQ > r, is said to lie outside the circle C (O, r) = {X, OX = r}
    The point outside the circle is also called exterior point.
    Point on the circle: A point S, such that OS = r is said to lie on the circle C(O, r) = {X ,OX = r}.
    Circular Disc: It is defined as a set of interior points and points on the circle. In set notation, it is written as : C(O, r) = {X : P OX ≤ r}
    What are the Parts of a Circle 12
  13. Concentric Circles:
    Circles having the same centre and different radius are said to be concentric circles.
    Remark. The word ‘radius’ is used for a line segment joining the centre to any point on the circle and also for its length.
    What are the Parts of a Circle 13
  14. Congruence of Circles & Arcs
    Congruent circles: Two circles are said to be congruent if and only if, one of them can be superposed on the other, so as the cover it exactly. It means two circles are congruent if and only if, their radii are equal. i.e., C (O, r) and C (O’ , r) are congruent if only if r = s.
    What are the Parts of a Circle 14Congruent arcs: Two arcs of a circle are congruent, if either of them can be superposed on the other, so as to cover it exactly. It is only possible, if degree measure of two arcs are the same.

Example 1: Take two points A and B on a plane sheet. Draw a circle with A as a centre, AC as radius and B in its exterior.
Solution: Mark two points A and B on a paper.
A •            • B
As the point B should be in the exterior of the circle, take A as the centre and radius (r) less than AB to draw a circle.
circle

Example 2 :Find the diameter of the circle of radius 6 cm.
Solution: We know,
Diameter = 2 × radius
∴ Diameter =2 × 6 cm =12 cm