Straight Lines

Circle

Conic Sections

Parabola

Ellipse

Hyperbola

Parametric Form of Conics

Asymptotes

Rectangular Hyperbola

Cartesian Plane

The horizontal line is called the x-axis, and the vertical line is called the y-axis. The coordinate axes divide the plane into four parts called quadrants. The point of intersection of the axes is called the origin.

The distance of a point from the y-axis is called its x-coordinate, or abscissa, and the distance of the point from the x-axis is called its y-coordinate, or ordinate.

Coordinates of Point

If the abscissa of a point is x and the ordinate is y, then (x, y) are called the coordinates of the point.

The coordinates of a point on the x-axis are of the form (x, 0) and that of the point on the y-axis are (0, y). The coordinates of the origin are (0, 0).

The coordinates of a point are of the form (+,+) in the first quadrant, (-,+) in the second quadrant, (-,-) in the third quadrant and (+,-) in the fourth quadrant.

Distance Formula

The distance between P(x1, y1) and Q(x2, y2) is

Mid-Point Formula

The mid-point of the line segment joining the points P(x1, y1) and Q(x2, y2) is

Area of Triangle

If A(x1, y1), B(x2 , y2) and C(x3, y3) be three vertices of a triangle ABC, then its area is given by

Colinear Points

Three points A, B and C are said to be collinear (lying on the same straight line) if:

  • AB + BC = AC or AC + CB = AB or AB + AC = BC
  • The area of the triangle formed by A, B and C is zero

Section Formula

If A and B are two points in a plane, then the coordinates of the point P which divides the line joining AB internally in the ratio m : n are

Straight Lines

A linear equation of the first degree in two variables x and y represents a straight line. The equation ax + by + c = 0 is the general form of equation of a line. Examples:

  • 2x + 3y + 6 = 0
  • 3y + 5 = 0 [a = 0]. The line is parallel to x-axis
  • 2x + 7 = 0 [b = 0]. The lines is parallel to y-axis
  • 2x + 3y = 0 [c = 0]. The line passes through the origin

The equation of x-axis is y = 0 and equation of y-axis is x = 0.

Slope of Line (m)

The slope is the tangent of the angle which the line makes with the x-axis, in the positive direction. You measure the angle from the x-axis towards the line, in the anti-clockwise direction.

The slope m is given as m = tan θ°. If θ is acute, the slope is positive and if θ is obtuse the slope is negative. 

The slope of a line parallel to x-axis is m = tan 0° = 0. The slope of a line perpendicular to x-axis is m = tan 90° = ∞.

Parallel Lines

Two lines are parallel to each other, if their slopes are equal. Thus, if the slope of line is m, the slope of a line parallel to it is also m. The equation of a line parallel to ax + by + c = 0 is ax + by = k.

Perpendicular Lines

Two lines are perpendicular to each other, if and only if the product of their slopes is -1. Thus, if the slope of a line is m, the slope of a line perpendicular to it is -1/m. The equation of a line perpendicular to ax + by + c = 0 is bx – ay = k.

Angle Between Two Lines

If the angle between two lines of slopes m1 and m2 be θ, then

tan θ = (m1 - m2)/(1 + m1m2)

Standard Forms of the Equation of a Line

Slope Intercept Form

The equation of a straight line having slope m and making an intercept c on y-axis is y = mx + c.

Two Point Form

The equation of a straight line passing through the points (x1, y1) and (x2, y2) is 

 

Point Slope Form

The equation of a straight line passing through the point (x1, y1) and having slope m is

(y – y1) = m(x – x1).

Double Intercept Form

The equation of a line making intercepts a and b on the x and y axes respectively is

x/a + y/b = 1.

The area of a triangle formed by the coordinate axes and the lines having intercepts a and b is ab/2.

Circles

The equation of circle whose centre is (h, k) and radius r is given by 

(x - h)2 + (y - k)2 = r2

The equation of circle whose centre is origin and radius r is

x2 + y2 = r2

General Equation

The equation x2 + y2 + 2gx + 2fy + c = 0 represents a circle of radius of √(g2 + f2 - c)

  • If g2 + f2 - c > 0, then the radius of the circle is real and hence the circle is also real.
  • If g2 + f2 - c = 0, then the radius of the circle is zero. Such a circle is known a point circle.
  • If g2 + f2 - c < 0, then the radius of circle is imaginary and such a circle is called an imaginary circle. 

Conic Sections

A conic section is the locus of a point which moves in such a way that its distance from a fixed point always bears a constant ratio to its distance from a fixed line. If S is a fixed point in the plane and ZZ´ is a fixed line in the same plane, then the locus of a point P which moves in the same plane in such a way that

SP/PM = Constant

Focus, Directrix, Ecentricity

The fixed point is called the focus of the conic section and the fixed line is known as its directrix. The constant ratio e is called the eccentricity of the conic section.

  • If e = 1, the conic is called a parabola
  • If e < 1, the conic is called an ellipse
  • If e > 1, the conic is called a hyperbola

General Equation of a Conic

Let S(α, β) be the focus, ax + by + c = 0 be the directrix and e be the eccentricity of a conic. Let P(h, k) be any point on the conic such that PM is the perpendicular from P on the directrix. Then, general equation is

The general equation of the second degree ax2 + 2hxy + by2 + 2gx + 2fy + c = 0 will represent

  • circle, if a = b and h = 0
  • parabola, if h2 = ab
  • ellipse, if h2 < ab
  • hyperbola, if h2 > ab
  • rectangular hyperbola, if h2 > ab and a + b = 0

Useful Terms

Axis: The straight line passing through the focus and perpendicular to the directrix is called the axis of the conic section.

Vertex: The point of intersection of the conic section and the axis is called the vertex of the conic section.

Centre: The point which bisects every chord of the conic passing through it, is called the centre of the conic section.

Latus-rectum: The latus rectum of a conic is the chord passing through the focus and perpendicular to the axis. 

Focal Chord: Any chord passing through the focus is called a focal chord of the conic section.

Double Ordinate: A chord perpendicular to the axis of a conic is known as its double ordinate.

Parabola

The standard equation of parabola is

y2 = 4ax 

(a, 0) is the focus of the parabola.

Parabola Types

Ellipse 

An ellipse is the focus of a point in a plane which moves in the plane in such a way that the ratio of its distance from a fixed point (called focus) in the same plane to its distance from a fixed straight line (called directrix) is always constant which is always less than unity. 

The equation of ellipse whose focus is (ae, 0) is

b2 = a2 (1 - e2)

Vertices: The point A and A', where the curve meets the line joining the foci S and S', are called the vertices of the ellipse. The coordinates of A and A' are (a, 0) and (–a, 0) respectively.

Major and Minor Axes

The distance AA' = 2a and BB' = 2b are called the major and minor axes of the ellipse respectively. Since, e < 1 and b2 = a2(1-e2). Therefore, a > b.

Foci

The points S(ae, 0) and S'(–ae, 0) are the foci of the ellipse.

Directrices

ZK and Z'K' are two directrices of the ellipse and their equations are x = a/e and x = - a/e respectively.

Hyperbola

A hyperbola is the locus of a point in a plane which moves in the plane in such a way that the ratio of its distance from a fixed point (called focus) in the same plane to its distance from fixed line (called directrix) is always constant which is always greater than unity.

Standard equation of hyperbola is:

 

b2 = a2 (e2 - 1)