Math Formula Sheet on Co-Ordinate Geometry

All grade math formula sheet on co-ordinate geometry. These math formula charts can be used by 10th grade, 11th grade, 12th grade and college grade students to solve co-ordinate geometry.

Co-Ordinate Geometry

● Rectangular Cartesian Co-ordinates:

(i) If the pole and initial line of the polar system coincides respectively with the origin and positive x-axis of the Cartesian system and (x, y), (r, θ) be the Cartesian and polar co-ordinates respectively of a point P on the plane then,

x = r cos θ, y = r sin θ

and r = √(x² + y²), θ = tan^-1(y/x).



(ii) The distance between two given points P (x₁, y₁) and Q (x₂, y₂) is

PQ = √{(x₂ - x₁)² + (y₂ - y₁)²}.

(iii) Let P (x₁, y₁) and Q (x₂, y₂) be two given points.

(a) If the point R divides the line-segment PQ internally in the ratio m : n, then the co-ordinates of R

are {(mx₂ + nx₁)/(m + n) , (my₂ + ny₁)/(m + n)}.

(b) If the point R divides the line-segment PQ externally in the ratio m : n, then the co-ordinates of R are

{(mx₂ - nx₁)/(m - n), (my₂ - ny₁)/(m - n)}.

(c) If R is the mid-point of the line-segment PQ, then the co-ordinates of R are {(x₁ + x₂)/2, (y₁ + y₂)/2}.

(iv) The co-ordinates of the centroid of the triangle formed by joining the points (x₁, y₁) , (x₂, y₂) and (x₃, y₃) are

({x₁ + x₂ + x₃}/3 , {y₁ + y₂ + y₃}/3

(v) The area of a triangle formed by joining the points (x₁, y₁), (x₂, y₂) and (x₃, y₃) is

½ | y₁ (x₂ - x₃) + y₂ (x₃ - x₁) + y₃ (x₁ - x₂) | sq. units

or, ½ | x₁ (y₂ - y₃) + x₂ (y₃ - y₁) + x₃ (y₁ - y₂) | sq. units.

● Straight Line:

(i) The slope or gradient of a straight line is the trigonometric tangent of the angle θ which the line makes with the positive directive of x-axis.

(ii) The slope of x-axis or of a line parallel to x-axis is zero.

(iii) The slope of y-axis or of a line parallel to y-axis is undefined.

(iv) The slope of the line joining the points (x₁, y₁) and (x₂, y₂) is

m = (y₂ - y₁)/(x₂ - x₁).

(v) The equation of x-axis is y = 0 and the equation of a line parallel to x-axis is y = b.

(vi) The equation of y-axis is x = 0 and the equation of a line parallel to y-axis is x = a.

(vii) The equation of a straight line in

(a) slope-intercept form: y = mx + c where m is the slope of the line and c is its y-intercept;

(b) point-slope form: y - y₁ = m (x - x₁) where m is the slope of the line and (x₁ , y₁) is a given point on the line;

(c) symmetrical form: (x - x₁)/cos θ = (y - y₁)/sin θ = r, where θ is the inclination of the line, (x₁, y₁) is a given point on the line and r is the distance between the points (x, y) and (x₁, y₁);

(d) two-point form: (x - x₁)/(x₂ - x₁) = (y - y₁)/(y₂ - y₁) where (x₁, y₁) and (x₂, y₂) are two given points on the line;

(e) intercept form: ^x/_a + ^y/_b = 1 where a = x-intercept and b = y-intercept of the line;

(f) normal form: x cos α + y sin α = p where p is the perpendicular distance of the line from the origin and α is the angle which the perpendicular line makes with the positive direction of the x-axis.

(g) general form: ax + by + c = 0 where a, b, c are constants and a, b are not both zero.

(viii) The equation of any straight line through the intersection of the lines a₁x + b₁y + c₁ = 0 and a₂x + b₂y + c₂ = 0 is a₁x + b₁y + c + k(a₂x + b₂y + c₂) = 0 (k ≠ 0).

(ix) If p ≠ 0, q ≠ 0, r ≠ 0 are constants then the lines a₁x + b₁y + c₁ = 0, a₂x + b₂y + c₂ = 0 and a₃x + b₃y + c₃ = 0 are concurrent if P(a₁x + b₁y + c₁) + q( a₂x + b₂y + c₂) + r(a₃x + b₃y + c₃) = 0.

(x) If θ be the angle between the lines y= m₁x + c₁ and y = m₂x + c₂ then tan θ = ± (m₁ - m₂ )/(1 + m₁ m₂);

(xi) The lines y= m₁x + c₁ and y = m₂x + c₂ are

(a) parallel to each other when m₁ = m₂;

(b) perpendicular to one another when m₁ ∙ m₂ = - 1.

(xii) The equation of any straight line which is

(a) parallel to the line ax + by + c = 0 is ax + by = k where k is an arbitrary constant;

(b) perpendicular to the line ax + by + c = 0 is bx - ay = k₁ where k₁ is an arbitrary constant.

(xiii) The straight lines a₁x + b₁y + c₁ = 0 and a₂x + b₂y + c₂ = 0 are identical if a₁/a₂ = b₁/b₂ = c₁/c₂.

(xiv) The points (x₁, y₁) and (x₂, y₂) lie on the same or opposite sides of the line ax + by + c = 0 according as (ax₁ + by₁ + c) and (ax₂ + by₂ + c) are of same sign or opposite signs.

(xv) Length of the perpendicular from the point (x1, y1) upon the line ax + by + c = 0 is|(ax₁ + by₁ + c)|/√(a² + b²).

(xvi) The equations of the bisectors of the angles between the lines a₁x + b₁y + c₁ = 0 and a₂x + b₂y + c₂ =0 are

(a₁x + b₁y + c₁)/√(a₁² + b₁²) = ± (a₂x + b₂y + c₂)/√(a₂² + b₂²).

● Circle:

(i) The equation of the circle having centre at the origin and radius a units is x² + y² = a² . . . (1)

The parametric equation of the circle (1) is x = a cos θ, y = a sin θ, θ being the parameter.

(ii) The equation of the circle having centre at (α, β) and radius a units is (x - α)² + (y - β)² = a².

(iii) The equation of the circle in general form is x² + y² + 2gx + 2fy + c = 0 The centre of this circle is at (-g, -f) and radius = √(g² + f² - c)

(iv) The equation ax² + 2hxy + by² + 2gx + 2fy + c = 0 represents a circle if a = b (≠ 0) and h = 0.

(v) The equation of a circle concentric with the circle x² + y² + 2gx + 2fy + c = 0 is x² + y² + 2gx + 2fy + k = 0 where k is an arbitrary constant.

(vi) If C₁ = x² + y² + 2g₁x + 2f₁y + c₁ = 0

and C₂ = x² + y² + 2g₂x + 2f₂y + c₂ = 0 then

(a) the equation of the circle passing through the points of intersection of C₁ and C₂ is C₁ + kC₂ = 0 (k ≠ 1);

(b) the equation of the common chord of C₁ and C₂ is C₁ - C₂ = 0.

(vii) The equation of the circle with the given points (x₁, y₁) and (x₂, y₂) as the ends of a diameter is (x - x₁) (x - x₂) + (y - y₁) (y - y₂) = 0.

(viii) The point (x₁, y₁) lies outside, on or inside the circle x² + y² + 2gx + 2fy + c = 0 according as x₁² + y₁² + 2gx₁ + 2fy₁ + c > , = or < 0.

● Parabola:

(i) Standard equation of parabola is y² = 4ax. Its vertex is the origin and axis is x-axis.

(ii) Other forms of the equations of parabola:

(a) x² = 4ay.

Its vertex is the origin and axis is y-axis.

(b) (y - β)² = 4a (x - α).

Its vertex is at (α, β) and axis is parallel to x-axis.

(c) (x - α)² = 4a(y- β).

Its vertex is at ( a, β) and axis is parallel to y-axis.

(iii) x = ay² + by + c (a ≠ o) represents equation of the parabola whose axis is parallel to x-axis.

(iv) y = px² + qx + r (p ≠ o) represents equation of the parabola whose axis is parallel to y-axis.

(v) The parametric equations of the parabola y² = 4ax are x = at² , y = 2at, t being the parameter.

(vi) The point (x₁, y₁) lies outside, on or inside the parabola y² = 4ax according as y₁² = 4ax₁ >, = or,<0

● Ellipse:

(i) Standard equation of ellipse is

x²/a² + y²/b² = 1 ……….(1)

(a) Its centre is the origin and major and minor axes are along x and y-axes respectively ; length of major axis = 2a and that of minor axis = 2b and eccentricity = e = √[1 – (b²/a²)]

(b) If S and S’ be the two foci and P (x, y) any point on it then SP = a - ex, S’P = a + ex and SP + S’P = 2a.

(c) The point (x₁, y₁) lies outside, on or inside the ellipse (1) according as x₁²/a² + y₁²/b² - 1 > , = or < 0.

(d) The parametric equations of the ellipse (1) are x = a cos θ, y = b sin θ where θ is the eccentric angle of the point P (x, y) on the ellipse (1) ; (a cos θ, b sin θ) are called the parametric co-ordinates of P.

(e) The equation of auxiliary circle of the ellipse (1) is x² + y² = a².

(ii) Other forms of the equations of ellipse:

(a) x²/a² + y²/b² = 1. Its centre is at the origin and the major and minor axes are along y and x-axes respectively.

(b) [(x - α)²]/a² + [(y - β)²]/b² = 1.

The centre of this ellipse is at (α, β) and the major and minor ones are parallel to x-axis and y-axis respectively.

● Hyperbola:

(i) Standard equation of hyperbola is x²/a² - y²/b² = 1 . . . (1)

(a) Its centre is the origin and transverse and conjugate axes are along x and y-axes respectively ; its length of transverse axis = 2a and that of conjugate axis = 2b and eccentricity = e = √[1 + (b²/a²)].

(b) If S and S’ be the two foci and P (x, y) any point on it then SP = ex - a, S’P = ex + a and S’P - SP = 2a.

(c) The point (x₁, y₁) lies outside, on or inside the hyperbola (1) according as x₁²/a² - y₁²/b² = -1 < , = or, > 0.

(d) The parametric equation of the hyperbola (1 ) are x = a sec θ, y = b tan θ and the parametric co-ordinates of any point P on (1) are (a sec θ,b tan θ).

(e) The equation of auxiliary circle of the hyperbola (1) is x² + y² = a².

(ii) Other forms of the equations of hyperbola:

(a) y²/a² - x²/b² = 1.

Its centre is the origin and transverse and conjugate axes are along y and x-axes respectively.

(b) [(x - α)²]/a² - [(y - β)²]/b² = 1. Its centre is at (α, β) and transverse and conjugate axes are parallel to x-axis and y-axis respectively.

(iii) Two hyperbolas
x²/a² - y²/b² = 1 ………..(2) and y²/b² - x²/a² = 1 …….. (3)

are conjugate to one another. If e₁ and e₂ be the eccentricities of the hyperbolas (2) and (3) respectively, then
b² = a² (e₁² - 1) and a² = b² (e₂² - 1).

(iv) The equation of rectangular hyperbola is x² - y² = a² ; its eccentricity = √2.

● Intersection of a Straight Line with a Conic:

(i) The equation of the chord of the

(a) circle x² + y² = a² which is bisected at (x₁, y₁) is T = S₁ where

T= xx₁ + yy₁ - a² and S₁ = x₁² - y₁² - a² ;

(b) circle x² + y² + 2gx + 2fy + c = 0 which is bisected at (x₁, y₁) is T = S₁ where T= xx₁ + yy₁ + g(x + x₁) + f(y + y₁) + c and S₁ = x₁² - y₁² + 2gx₁ +2fy₁ + c;

(c) parabola y² = 4ax which is bisected at (x₁,y₁) is T = S₁ where T = yy₁ - 2a (x + x₁) and S₁ = y₁² - 4ax₁;

(d) ellipse x²/a² + y²/b² = 1 which is bisected at (x₁,y₁) is T = S₁

where T = (xx₁)/a² + (yy₁)/b² - 1 and S₁ = x₁²/a² + y₁²/b² - 1.

(e) hyperbola x²/a² - y²/b² = 1 which is bisected at (x₁, y₁) is T = S₁

where T = {(xx₁)/a²} – {(yy₁)/b²} - 1 and S₁ = (x₁²/a²) + (y₁²/b²) - 1.

(ii) The equation of the diameter of a conic which bisects all chords parallel to the line y = mx + c is

(a) x + my = 0 when the conic is the circle x² + y² = a² ;

(b) y = 2a/m when the conic is the parabola y² = 4ax;

(c) y = - [b²/(a²m)] ∙ x when the conic is the ellipse x²/a² + y²/b² = 1

(d) y = [b²/(a²m )] ∙ x when the conic is the hyperbola x²/a² - y²/b² = 1

(iii) y = mx and y = m’x are two conjugate diameters of the

(a) ellipse x²/a² + y²/b² = 1 when mm’ = - b²/a²

(b) hyperbola x²/a² - y²/b² = 1 when mm’ = b²/a².

● Formula

• Basic Math Formulas
  
• Math Formula Sheet on Co-Ordinate Geometry
  
• All Math Formula on Mensuration
  
• Simple Math Formula on Trigonometry

11 and 12 Grade Math

From Math Formula Sheet on Co-Ordinate Geometry to HOME PAGE