Co-ordinate Geometry

Welcome

Welcome to Co-ordinate Geometry, this app has all the important co-ordinate geometry formulas requried at high school level.

Sign Convention
Straight Lines
Pair Of Straight Lines
Circles
Parabola
Ellipse
Hyperbola

About this app

Quadrants and sign of co-ordinates:
1. In first quadrant, (+, +)
2. In second quadrant, (-, +)
3. In third quadrant, (-, -)
4. In forth quadrant, (+, -)
Where first sign is for x-co-ordinate and second sign for y-co-ordinate

1.   Distance between the line Ax + By + c = 0 and the given ordinate
$Description: \left(x_1, y_1 \right)$ is: $Description: d = \dfrac{Ax_1 + By_1 + C}{\sqrt{A^2 + B^2}}$
2.   The distance between any two points  and  is given by $Description: AB = d = \sqrt{ \left( x_2- x_1 \right) ^2 + \left( y_2- y_1 \right) ^2}$
3.   The distance between any two points  and origin (0, 0) is given by $Description: (d) = \sqrt{(x^2 + y^2)}$
4.   The co-ordinate of the point which divides a straight line joining the given points internally in the ratio  is given by
$Description: \therefore X = \dfrac{m_1x_2- m_2x_1}{m_2- m_1}, \, Y = \dfrac{m_1y_2- m_2y_1}{m_1- m_2}$ Or,
$Description: \therefore (X, Y) = \left(\dfrac{m_1x_2- m_2x_1}{m_2- m_1}, \dfrac{m_1y_1- m_2y_1}{m_1- m_2}\right)$ is section formula.
5.   The co-ordinate of the point which bisects the line joining two given points  and is given by
$Description: X = \dfrac{x_1 + x_2}{2}$ and, $Description: Y = \dfrac{y_1 + y_2}{2}$
If the point divides AB in the ratio k:1 then
$Description: X =\dfrac{kx_2 + x_1}{k + 1}$ and, $Description: Y = \dfrac{ky_2 + y_1}{k + 1}$
6.   The co-ordinate of the centroid of a triangle ABC whose vertices are  and  is the given by
$Description: X = \dfrac{x_1 + x_2 + x_3}{3}$ and $Description: Y = \dfrac{y_1 + y_2 + y_3}{3}$
7.   a. the area $Description: \Delta$ ABC whose vertices are
and  is given by
$Description: \Delta ABC = \dfrac{1}{2} \begin{pmatrix}x_1x_2 & x_3x_1 \\ y_1y_3 & y_3y_1\end{pmatrix}$
= $Description: \dfrac{1}{2}[(x_1y_2- x_2y_1) + (x_2y_3- x_3y_2) + (x_3y_1- x_1y_3)]$
b. if one of the vertices of $Description: \Delta ABC$ is the origin
$Description: \Delta ABC = (x_1y_2- x_2y_1)$

8.    let  and  be the vertices of a quadrilateral. Then the area of quadrilateral ABCD is
$Description: A = \dfrac{1}{2}[(x_1y_2- x_2y_1) +( x_2y_3- x_3y_2) + (x_3y_4- x_4y_3) + (x_4y_1- x_1y_4)]$
EQUATION OF STRAIGHT LINES
1.   the slope of the line joining two points  and  is given by,
a.   slope of the line $Description: m = (\tan\theta) = \dfrac{y_2- y_1}{x_2- x_1}$
b.   when the equation of line is given, (ax + by – c = 0) then slope of line = $Description: \dfrac{-a}{b} = -\dfrac{coefficient of x}{coefficient of y}$
2.   equation of straight line in slope & intercept form is y = mx + c where m is slope and c is x-intercept
3.   when the line passes through the origin, then c = 0 & equation is y = mx.
4.   When the line is parallel to x-axis, then the equation is y = h.
5.   When the line is parallel to y-axis then the equation is x = a.
6.   Equation of straight line in double intercept form is, $Description: \dfrac{x}{a} + \dfrac{y}{b} = 1$
7.   Equation of straight line in normal (perpendicular form is, $Description: x\cos\alpha + y\sin\alpha = P$ .
8.   Equation of a line passing from two points  and  is $Description: (y- y_1) = \dfrac{y_2- y_1}{x_2- x_1}(x- x_1)$
9.   Equation of line passing from a point  & slope ‘m’ is,
10.   The angle between two lines  and  is $Description: \theta = \tan^{-1}(\dfrac{m_1- m_2}{1 + m_1m_2})$ where  = slope of first line &  = slope of second line.
11.   When two lines are parallel, then  (their slope is equal)
12.   When tow lines are perpendicular, then .
13.   The angle between two lines $Description: A_1x + B_1y + c_1 = 0 \, \& \, Ax_2 + By_2 + c_2 = 0$ is, $Description: \theta = \tan^{-1}(\pm\dfrac{A_2B_1- A_1B_2}{A_1A_2 + B_1B_2})$

1.   The general equation of any line through the intersection of two lines  and  is  where k is given by  and  is the point of intersection of the two lines.
2.   If  represents the equation of a pair of straight lines through the origin, then $Description: ax + (h + \sqrt{h^2- ab})y = 0, ax + (h- \sqrt{(h^2- ab)}y = 0$ are the equation of two straight lines represented by the homogeneous equation of the second degree.
3.   The angle between a pair of lines represented by the equation  is given by, $Description: \tan \theta = \pm\dfrac{2\sqrt{h^2- ab}}{a + b}$ if $Description: h^2 = ab, \theta = 0 ^\circ$ i.e. the straight lines are coincident. If $Description: a + b = 0, \theta = 90^\circ$ i.e. the straight lines are perpendicular.

1. Standard equation of a circle

(x-h)²+(y-k)² = a²

Centre of the circle is at (h,k)
radius of the circule is a

2. Some particular cases of standard equation of a circle
i) Centre is at origin h = 0, and k = 0

x²+y² = a²

(ii) Circle passes through origin
So radius = a² = h²+k²

(x-h)²+(y-k)² = h²+k²

(iii)Circle touches the x axis
C(h,k) centre, a = radius
To satisfy a = k
So equation is
(x-h)²+(y-a)² = a²

(iv)Circle touches the y axis
C(h,k) centre, a = radius
To satisfy a = h
So equation is
(x-a)²+(y-k)² = a²

(v) When the circle touches both axes

then h = k = a
(x-a)²+(y-a)² = a²

(vi) When the circle passes through the origin and centre is on x-axis.
C(h,k) centre, a = radius

As centre is on x axis y coordinate is zero. So k = 0.
As circle is passing through origin a = h
(x-a)²+ y² = a²

(vii) When the circle passes through the origin and centre is on y-axis.
C(h,k) centre, a = radius

As centre is on y axis x coordinate is zero. So h = 0.
As circle is passing through origin a = k
x²+(y-a)² = a²

3. General equation of a circle

x²+y²+2gx+2fy+c = 0

Centre of this circle = (-g,-f)
Radius = √(g²+f²-c)

4. Equation of a circle when the coordinates of end points of a diameter are given

If (x1,y1) and (x2,y2) are coordinates of end points of the diameter

then the equation of the circle is
(x - x1)(x - x2)+(y - y1)(y- y2) = o

The equation of a parabola can be written in two basic forms:

Form 1: y = a( x – h)2 + k
Form 2: x = a( y – k)2 + h

In Form 1, the parabola opens vertically. (It opens in the “ y” direction.) If a > 0, it opens upward. Refer to Figure 1(a). If a < 0, it opens downward. The distance from the vertex to the focus and from the vertex to the directrix line are the same. This distance is

A parabola with its vertex at ( h, k), opening vertically, will have the following properties.

The focus will be at .
The directrix will have the equation .
The axis of symmetry will have the equation x = h.
Its form will be y = a( x – h)2 + k.

In Form 2, the parabola opens horizontally. (It opens in the “ x” direction.) If a > 0, it opens to the right. Refer to Figure 1(b). If a < 0, it opens to the left.
A parabola with its vertex at ( h, k), opening horizontally, will have the following properties.

The focus will be at .
The directrix will have the equation .
The axis of symmetry will have the equation y = k.
Its form will be x = a( y – k)2 + h

Horizontal:-
Description: http://www.clausentech.com/lchs/dclausen/algebra2/lecture_notes/conics/hyperb1.gif
"a" is the number in the denominator of the positive term
If the x-term is positive, then the hyperbola is horizontal
a = semi-transverse axis
b = semi-conjugate axis
center: (h, k)
vertices: (h + a, k), (h - a, k)
c = distance from the center to each focus along the transverse axis

foci: (h + c, k), (h - c, k)

The eccentricity e > 1
Description: http://www.clausentech.com/lchs/dclausen/algebra2/lecture_notes/conics/hyperb6.gif

Vertical-:
Description: http://www.clausentech.com/lchs/dclausen/algebra2/lecture_notes/conics/hyperb3.gif
"a" is the number in the denominator of the positive term
If the y-term is positive, then hyperbola is vertical
a = semi-transverse axis
b = semi-conjugate axis
center: (h, k)
vertices: (h, k + a), (h, k - a)
c = distance from the center to each focus along the transverse axis

foci: (h, k + c), (h, k - c)

The eccentricity e > 1

Description: http://www.clausentech.com/lchs/dclausen/algebra2/lecture_notes/conics/hyperb5.gif

Horizontal:
Description: http://www.clausentech.com/lchs/dclausen/algebra2/lecture_notes/conics/ellips4.gif
a2 > b2
If the larger denominator is under the "x" term, then the ellipse is horizontal.
center (h, k)
a = length of semi-major axis
b = length of semi-minor axis
vertices: (h + a, k), (h - a, k)
co-vertices: (h, k + b), (h, k - b) [endpoints of the minor axis]
c is the distance from the center to each focus.

foci: (h + c, k), (h - c, k)
0 < e < 1 for an ellipse…e is eccentricity

Vertical:
Description: http://www.clausentech.com/lchs/dclausen/algebra2/lecture_notes/conics/vert_ellipse_eq.jpg
a2 > b2
If the larger denominator is under the "y" term, then the ellipse is vertical.
center (h, k)
a = length of semi-major axis
b = length of semi-minor axis
vertices: (h, k + a), (h, k - a)
co-vertices: (h + b, k), (h - b, k) [endpoints of the minor axis]
c is the distance from the center to each focus.

foci: (h, k + c), (h, k - c)

0 < e < 1 for an ellipse…e is eccentricity

App developed by Team Helium.

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Thanks for installing.