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, ?) be the Cartesian and polar co-ordinates respectively of a point P on the plane then,
x = r cos ?, y = r sin ?
and r = ?(x2 + y2), ? = 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
1/2| y1 (x2 - x3) + y2 (x3 - x1) + y3 (x1 - x2) | sq. units
or, 1/2 | 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 ? 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 ? = (y - y1)/sin ? = r, where ? 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 ? + y sin ? = p where p is the perpendicular distance of the line from the origin and ? 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 ? be the angle between the lines y= m1x + c1 and y = m2x + c2 then tan ? = ± (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 ?, y = a sin ?, ? being the parameter.

(ii) The equation of the circle having centre at (?, ?) and radius a units is (x - ?)2 + (y - ?)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 - ?)2 = 4a (x - ?).
Its vertex is at (?, ?) and axis is parallel to x-axis.
(c) (x - ?)2 = 4a(y- ?).
Its vertex is at ( a, ?) 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 ?, y = b sin ? where ? is the eccentric angle of the point P (x, y) on the ellipse (1) ; (a cos ?, b sin ?) 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 - ?)2]/a2 + [(y - ?)2]/b2 = 1.
The centre of this ellipse is at (?, ?) 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 ?, y = b tan ? and the parametric co-ordinates of any point P on (1) are (a sec ?,b tan ?).
(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 - ?)2]/a2 - [(y - ?)2]/b2 = 1. Its centre is at (?, ?) 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.