A coordinate geometry is a branch of geometry where the position of the points on the plane is defined with the help of an ordered pair of numbers also known as coordinates.

1. Straight Lines

  1. Equation of Straight Line
  2. Parametric Form of Straight Line
  3. Perpendicular Distance From Point to Straight Line
  4. Angle Between Two Straight Lines
  5. Condition for Three Lines To Be Concurrent
  6. Pair of Straight Lines

2. Circle

  1. Equation of a Circle
  2. General Equation of Circle
  3. Parametric Form of Circle
  4. Equation of Tangent to a Circle
  5. Length of Tangent to Circle
  6. Condition For Line To Be Tangent To Circle
  7. Equation of Chord of Contact of Tangents
  8. Circles Touching Each Other
  9. Orthogonal Circles

3. Conic Sections

  1. Definition of a Conic
  2. General Equation of a Conic
  3. Classification of Conics

Parabola

  1. Standard Equation of Parabola
  2. Important Definitions Regarding Parabola
  3. Standard Parabolas

Ellipse

  1. Standard Equation of Ellipse
  2. Important Definitions Regarding Ellipse
  3. Other Standard Form of Ellipse

Hyperbola

  1. Standard Equation of Hyperbola
  2. Important Definitions Regarding Hyperbola
  3. Other Form of 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)