Math Formula Sheet on CoOrdinate Geometry
All grade math formula sheet on coordinate geometry. These math formula charts can be used by 10th grade, 11th grade, 12th grade and college grade students to solve coordinate geometry.
CoOrdinate Geometry
● Rectangular Cartesian Coordinates:
(i) If the pole and initial line of the polar system coincides respectively with the origin and positive xaxis of the Cartesian system and (x, y), (r, θ) be the Cartesian and polar coordinates respectively of a point P on the plane then,
x = r cos θ, y = r sin θ
and r = √(x
^{2} + y
^{2}), θ = tan
^{1}(y/x).
(ii) The distance between two given points P (x
_{1}, y
_{1}) and Q (x
_{2}, y
_{2}) is
PQ = √{(x
_{2}  x
_{1})
^{2} + (y
_{2}  y
_{1})
^{2}}.
(iii) Let P (x
_{1}, y
_{1}) and Q (x
_{2}, y
_{2}) be two given points.
(a) If the point R divides the linesegment
PQ internally in the ratio m : n, then the coordinates of R
are {(mx
_{2} + nx
_{1})/(m + n) , (my
_{2} + ny
_{1})/(m + n)}.
(b) If the point R divides the linesegment
PQ externally in the ratio m : n, then the coordinates of R are
{(mx
_{2}  nx
_{1})/(m  n), (my
_{2}  ny
_{1})/(m  n)}.
(c) If R is the midpoint of the linesegment
PQ, then the coordinates of R are {(x
_{1} + x
_{2})/2, (y
_{1} + y
_{2})/2}.
(iv) The coordinates of the centroid of the triangle formed by joining the points (x
_{1}, y
_{1}) , (x
_{2}, y
_{2}) and (x
_{3}, y
_{3}) are
({x
_{1} + x
_{2} + x
_{3}}/3 , {y
_{1} + y
_{2} + y
_{3}}/3
(v) The area of a triangle formed by joining the points (x
_{1}, y
_{1}), (x
_{2}, y
_{2}) and (x
_{3}, y
_{3}) is
½  y
_{1} (x
_{2}  x
_{3}) + y
_{2} (x
_{3}  x
_{1}) + y
_{3} (x
_{1}  x
_{2})  sq. units
or, ½  x
_{1} (y
_{2}  y
_{3}) + x
_{2} (y
_{3}  y
_{1}) + x
_{3} (y
_{1}  y
_{2})  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 xaxis.
(ii) The slope of xaxis or of a line parallel to xaxis is zero.
(iii) The slope of yaxis or of a line parallel to yaxis is undefined.
(iv) The slope of the line joining the points (x
_{1}, y
_{1}) and (x
_{2}, y
_{2}) is
m = (y
_{2}  y
_{1})/(x
_{2}  x
_{1}).
(v) The equation of xaxis is y = 0 and the equation of a line parallel to xaxis is y = b.
(vi) The equation of yaxis is x = 0 and the equation of a line parallel to yaxis is x = a.
(vii) The equation of a straight line in
(a) slopeintercept form: y = mx + c where m is the slope of the line and c is its yintercept;
(b) pointslope form: y  y
_{1} = m (x  x
_{1}) where m is the slope of the line and (x
_{1} , y
_{1}) is a given point on the line;
(c) symmetrical form: (x  x
_{1})/cos θ = (y  y
_{1})/sin θ = r, where θ is the inclination of the line, (x
_{1}, y
_{1}) is a given point on the line and r is the distance between the points (x, y) and (x
_{1}, y
_{1});
(d) twopoint form: (x  x
_{1})/(x
_{2}  x
_{1}) = (y  y
_{1})/(y
_{2}  y
_{1}) where (x
_{1}, y
_{1}) and (x
_{2}, y
_{2}) are two given points on the line;
(e) intercept form:
^{x}/
_{a} +
^{y}/
_{b} = 1 where a = xintercept and b = yintercept 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 xaxis.
(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 a
_{1}x + b
_{1}y + c
_{1} = 0 and a
_{2}x + b
_{2}y + c
_{2} = 0 is a
_{1}x + b
_{1}y + c + k(a
_{2}x + b
_{2}y + c
_{2}) = 0 (k ≠ 0).
(ix) If p ≠ 0, q ≠ 0, r ≠ 0 are constants then the lines a
_{1}x + b
_{1}y + c
_{1} = 0, a
_{2}x + b
_{2}y + c
_{2} = 0 and a
_{3}x + b
_{3}y + c
_{3} = 0 are concurrent if P(a
_{1}x + b
_{1}y + c
_{1}) + q( a
_{2}x + b
_{2}y + c
_{2}) + r(a
_{3}x + b
_{3}y + c
_{3}) = 0.
(x) If θ be the angle between the lines y= m
_{1}x + c
_{1} and y = m
_{2}x + c
_{2} then tan θ = ± (m
_{1}  m
_{2} )/(1 + m
_{1} m
_{2});
(xi) The lines y= m
_{1}x + c
_{1} and y = m
_{2}x + c
_{2} are
(a) parallel to each other when m
_{1} = m
_{2};
(b) perpendicular to one another when m
_{1} ∙ m
_{2} =  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 = k
_{1} where k
_{1} is an arbitrary constant.
(xiii) The straight lines a
_{1}x + b
_{1}y + c
_{1} = 0 and a
_{2}x + b
_{2}y + c
_{2} = 0 are identical if a
_{1}/a
_{2} = b
_{1}/b
_{2} = c
_{1}/c
_{2}.
(xiv) The points (x
_{1}, y
_{1}) and (x
_{2}, y
_{2}) lie on the same or opposite sides of the line ax + by + c = 0 according as (ax
_{1} + by
_{1} + c) and (ax
_{2} + by
_{2} + 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(ax
_{1} + by
_{1} + c)/√(a
^{2} + b
^{2}).
(xvi) The equations of the bisectors of the angles between the lines a
_{1}x + b
_{1}y + c
_{1} = 0 and a
_{2}x + b
_{2}y + c
_{2} =0 are
(a
_{1}x + b
_{1}y + c
_{1})/√(a
_{1}^{2} + b
_{1}^{2}) = ± (a
_{2}x + b
_{2}y + c
_{2})/√(a
_{2}^{2} + b
_{2}^{2}).
● Circle:
(i) The equation of the circle having centre at the origin and radius a units is x
^{2} + y
^{2} = a
^{2} . . . (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} = a
^{2}.
(iii) The equation of the circle in general form is x
^{2} + y
^{2} + 2gx + 2fy + c = 0 The centre of this circle is at (g, f) and radius = √(g
^{2} + f
^{2}  c)
(iv) The equation ax
^{2} + 2hxy + by
^{2} + 2gx + 2fy + c = 0 represents a circle if a = b (≠ 0) and h = 0.
(v) The equation of a circle concentric with the circle x
^{2} + y
^{2} + 2gx + 2fy + c = 0 is x
^{2} + y
^{2} + 2gx + 2fy + k = 0 where k is an arbitrary constant.
(vi) If C
_{1} = x
^{2} + y
^{2} + 2g
_{1}x + 2f
_{1}y + c
_{1} = 0
and C
_{2} = x
^{2} + y
^{2} + 2g
_{2}x + 2f
_{2}y + c
_{2} = 0 then
(a) the equation of the circle passing through the points of intersection of C
_{1} and C
_{2} is C
_{1} + kC
_{2} = 0 (k ≠ 1);
(b) the equation of the common chord of C
_{1} and C
_{2} is C
_{1}  C
_{2} = 0.
(vii) The equation of the circle with the given points (x
_{1}, y
_{1}) and (x
_{2}, y
_{2}) as the ends of a diameter is
(x  x
_{1}) (x  x
_{2}) + (y  y
_{1}) (y  y
_{2}) = 0.
(viii) The point (x
_{1}, y
_{1}) lies outside, on or inside the circle x
^{2} + y
^{2} + 2gx + 2fy + c = 0 according as x
_{1}^{2} + y
_{1}^{2} + 2gx
_{1} + 2fy
_{1} + c > , = or < 0.
● Parabola:
(i) Standard equation of parabola is y
^{2} = 4ax. Its vertex is the origin and axis is xaxis.
(ii) Other forms of the equations of parabola:
(a) x
^{2} = 4ay.
Its vertex is the origin and axis is yaxis.
(b) (y  β)
^{2} = 4a (x  α).
Its vertex is at (α, β) and axis is parallel to xaxis.
(c) (x  α)
^{2} = 4a(y β).
Its vertex is at ( a, β) and axis is parallel to yaxis.
(iii) x = ay
^{2} + by + c (a ≠ o) represents equation of the parabola whose axis is parallel to xaxis.
(iv) y = px
^{2} + qx + r (p ≠ o) represents equation of the parabola whose axis is parallel to yaxis.
(v) The parametric equations of the parabola y
^{2} = 4ax are x = at
^{2} , y = 2at, t being the parameter.
(vi) The point (x
_{1}, y
_{1}) lies outside, on or inside the parabola y
^{2} = 4ax according as y
_{1}^{2} = 4ax
_{1} >, = or,<0
● Ellipse:
(i) Standard equation of ellipse is
x
^{2}/a
^{2} + y
^{2}/b
^{2} = 1 ……….(1)
(a) Its centre is the origin and major and minor axes are along x and yaxes respectively ; length of major axis = 2a and that of minor axis = 2b and eccentricity = e = √[1 – (b
^{2}/a
^{2})]
(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 (x
_{1}, y
_{1}) lies outside, on or inside the ellipse (1) according as x
_{1}^{2}/a
^{2} + y
_{1}^{2}/b
^{2}  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 coordinates of P.
(e) The equation of auxiliary circle of the ellipse (1) is x
^{2} + y
^{2} = a
^{2}.
(ii) Other forms of the equations of ellipse:
(a) x
^{2}/a
^{2} + y
^{2}/b
^{2} = 1. Its centre is at the origin and the major and minor axes are along y and xaxes respectively.
(b) [(x  α)
^{2}]/a
^{2} + [(y  β)
^{2}]/b
^{2} = 1.
The centre of this ellipse is at (α, β) and the major and minor ones are parallel to xaxis and yaxis respectively.
● Hyperbola:
(i) Standard equation of hyperbola is x
^{2}/a
^{2}  y
^{2}/b
^{2} = 1 . . . (1)
(a) Its centre is the origin and transverse and conjugate axes are along x and yaxes respectively ; its length of transverse axis = 2a and that of conjugate axis = 2b and eccentricity = e = √[1 + (b
^{2}/a
^{2})].
(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 (x
_{1}, y
_{1}) lies outside, on or inside the hyperbola (1) according as x
_{1}^{2}/a
^{2}  y
_{1}^{2}/b
^{2} = 1 < , = or, > 0.
(d) The parametric equation of the hyperbola (1 ) are x = a sec θ, y = b tan θ and the parametric coordinates of any point P on (1) are (a sec θ,b tan θ).
(e) The equation of auxiliary circle of the hyperbola (1) is x
^{2} + y
^{2} = a
^{2}.
(ii) Other forms of the equations of hyperbola:
(a) y
^{2}/a
^{2}  x
^{2}/b
^{2} = 1.
Its centre is the origin and transverse and conjugate axes are along y and xaxes respectively.
(b) [(x  α)
^{2}]/a
^{2}  [(y  β)
^{2}]/b
^{2} = 1. Its centre is at (α, β) and transverse and conjugate axes are parallel to xaxis and yaxis respectively.
(iii) Two hyperbolas
x
^{2}/a
^{2}  y
^{2}/b
^{2} = 1 ………..(2) and y
^{2}/b
^{2}  x
^{2}/a
^{2} = 1 …….. (3)
are conjugate to one another. If e
_{1} and e
_{2} be the eccentricities of the hyperbolas (2) and (3) respectively, then
b
^{2} = a
^{2} (e
_{1}^{2}  1) and a
^{2} = b
^{2} (e
_{2}^{2}  1).
(iv) The equation of rectangular hyperbola is x
^{2}  y
^{2} = a
^{2} ; its eccentricity = √2.
● Intersection of a Straight Line with a Conic:
(i) The equation of the chord of the
(a) circle x
^{2} + y
^{2} = a
^{2} which is bisected at (x
_{1}, y
_{1}) is T = S
_{1} where
T= xx
_{1} + yy
_{1}  a
^{2} and S
_{1} = x
_{1}^{2}  y
_{1}^{2}  a
^{2} ;
(b) circle x
^{2} + y
^{2} + 2gx + 2fy + c = 0 which is bisected at (x
_{1}, y
_{1}) is T = S
_{1} where T= xx
_{1} + yy
_{1} + g(x + x
_{1}) + f(y + y
_{1}) + c and S
_{1} = x
_{1}^{2}  y
_{1}^{2} + 2gx
_{1} +2fy
_{1} + c;
(c) parabola y
^{2} = 4ax which is bisected at (x
_{1},y
_{1}) is T = S
_{1} where T = yy
_{1}  2a (x + x
_{1}) and S
_{1} = y
_{1}^{2}  4ax
_{1};
(d) ellipse x
^{2}/a
^{2} + y
^{2}/b
^{2} = 1 which is bisected at (x
_{1},y
_{1}) is T = S
_{1} where T = (xx
_{1})/a
^{2} + (yy
_{1})/b
^{2}  1 and S
_{1} = x
_{1}^{2}/a
^{2} + y
_{1}^{2}/b
^{2}  1.
(e) hyperbola x
^{2}/a
^{2}  y
^{2}/b
^{2} = 1 which is bisected at (x
_{1}, y
_{1}) is T = S
_{1} where T = {(xx
_{1})/a
^{2}} – {(yy
_{1})/b
^{2}}  1 and S
_{1} = (x
_{1}^{2}/a
^{2}) + (y
_{1}^{2}/b
^{2})  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 x
^{2} + y
^{2} = a
^{2} ;
(b) y = 2a/m when the conic is the parabola y
^{2} = 4ax;
(c) y =  [b
^{2}/(a
^{2}m)] ∙ x when the conic is the ellipse x
^{2}/a
^{2} + y
^{2}/b
^{2} = 1
(d) y = [b
^{2}/(a
^{2}m )] ∙ x when the conic is the hyperbola x
^{2}/a
^{2}  y
^{2}/b
^{2} = 1
(iii) y = mx and y = m’x are two conjugate diameters of the
(a) ellipse x
^{2}/a
^{2} + y
^{2}/b
^{2} = 1 when mm’ =  b
^{2}/a
^{2} (b) hyperbola x
^{2}/a
^{2}  y
^{2}/b
^{2} = 1 when mm’ = b
^{2}/a
^{2}.
● Formula
11 and 12 Grade Math
From Math Formula Sheet on CoOrdinate Geometry to HOME PAGE
Didn't find what you were looking for? Or want to know more information
about Math Only Math.
Use this Google Search to find what you need.
Share this page:
What’s this?

