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 = √(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 line-segment
PQ internally in the ratio m : n, then the co-ordinates of R
are {(mx
_{2} + nx
_{1})/(m + n) , (my
_{2} + ny
_{1})/(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
{(mx
_{2} - nx
_{1})/(m - n), (my
_{2} - ny
_{1})/(m - n)}.
(c) If R is the mid-point of the line-segment
PQ, then the co-ordinates of R are {(x
_{1} + x
_{2})/2, (y
_{1} + y
_{2})/2}.
(iv) The co-ordinates 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 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 (x
_{1}, y
_{1}) and (x
_{2}, y
_{2}) is
m = (y
_{2} - y
_{1})/(x
_{2} - x
_{1}).
(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 - 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) two-point 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 = 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 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 x-axis.
(ii) Other forms of the equations of parabola:
(a) x
^{2} = 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 = ay
^{2} + by + c (a ≠ o) represents equation of the parabola whose axis is parallel to x-axis.
(iv) y = px
^{2} + qx + r (p ≠ o) represents equation of the parabola whose axis is parallel to y-axis.
(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 y-axes 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 co-ordinates 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 x-axes 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 x-axis and y-axis 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 y-axes 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 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 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 x-axes 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 x-axis and y-axis 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
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