A coordinate geometry is a branch of geometry where the position of the points on the plane is defined with the help of an ordered pair of numbers also known as coordinates.
The horizontal line is called the x-axis, and the vertical line is called the y-axis. The coordinate axes divide the plane into four parts called quadrants. The point of intersection of the axes is called the origin.
The distance of a point from the y-axis is called its x-coordinate, or abscissa, and the distance of the point from the x-axis is called its y-coordinate, or ordinate.
Coordinates of Point
If the abscissa of a point is x and the ordinate is y, then (x, y) are called the coordinates of the point.
The coordinates of a point on the x-axis are of the form (x, 0) and that of the point on the y-axis are (0, y). The coordinates of the origin are (0, 0).
The coordinates of a point are of the form (+,+) in the first quadrant, (-,+) in the second quadrant, (-,-) in the third quadrant and (+,-) in the fourth quadrant.
The distance between P(x1, y1) and Q(x2, y2) is
The mid-point of the line segment joining the points P(x1, y1) and Q(x2, y2) is
Area of Triangle
If A(x1, y1), B(x2 , y2) and C(x3, y3) be three vertices of a triangle ABC, then its area is given by
Three points A, B and C are said to be collinear (lying on the same straight line) if:
If A and B are two points in a plane, then the coordinates of the point P which divides the line joining AB internally in the ratio m : n are
A linear equation of the first degree in two variables x and y represents a straight line. The equation ax + by + c = 0 is the general form of equation of a line. Examples:
The equation of x-axis is y = 0 and equation of y-axis is x = 0.
Slope of Line (m)
The slope is the tangent of the angle which the line makes with the x-axis, in the positive direction. You measure the angle from the x-axis towards the line, in the anti-clockwise direction.
The slope m is given as m = tan θ°. If θ is acute, the slope is positive and if θ is obtuse the slope is negative.
The slope of a line parallel to x-axis is m = tan 0° = 0. The slope of a line perpendicular to x-axis is m = tan 90° = ∞.
Two lines are parallel to each other, if their slopes are equal. Thus, if the slope of line is m, the slope of a line parallel to it is also m. The equation of a line parallel to ax + by + c = 0 is ax + by = k.
Two lines are perpendicular to each other, if and only if the product of their slopes is -1. Thus, if the slope of a line is m, the slope of a line perpendicular to it is -1/m. The equation of a line perpendicular to ax + by + c = 0 is bx – ay = k.
Angle Between Two Lines
If the angle between two lines of slopes m1 and m2 be θ, then
tan θ = (m1 - m2)/(1 + m1m2)
Slope Intercept Form
The equation of a straight line having slope m and making an intercept c on y-axis is y = mx + c.
Two Point Form
The equation of a straight line passing through the points (x1, y1) and (x2, y2) is
Point Slope Form
The equation of a straight line passing through the point (x1, y1) and having slope m is
(y – y1) = m(x – x1).
Double Intercept Form
The equation of a line making intercepts a and b on the x and y axes respectively is
x/a + y/b = 1.
The area of a triangle formed by the coordinate axes and the lines having intercepts a and b is ab/2.
The equation of circle whose centre is (h, k) and radius r is given by
(x - h)2 + (y - k)2 = r2
The equation of circle whose centre is origin and radius r is
x2 + y2 = r2
The equation x2 + y2 + 2gx + 2fy + c = 0 represents a circle of radius of √(g2 + f2 - c)
A conic section is the locus of a point which moves in such a way that its distance from a fixed point always bears a constant ratio to its distance from a fixed line. If S is a fixed point in the plane and ZZ´ is a fixed line in the same plane, then the locus of a point P which moves in the same plane in such a way that
SP/PM = Constant
Focus, Directrix, Ecentricity
The fixed point is called the focus of the conic section and the fixed line is known as its directrix. The constant ratio e is called the eccentricity of the conic section.
General Equation of a Conic
Let S(α, β) be the focus, ax + by + c = 0 be the directrix and e be the eccentricity of a conic. Let P(h, k) be any point on the conic such that PM is the perpendicular from P on the directrix. Then, general equation is
The general equation of the second degree ax2 + 2hxy + by2 + 2gx + 2fy + c = 0 will represent
Axis: The straight line passing through the focus and perpendicular to the directrix is called the axis of the conic section.
Vertex: The point of intersection of the conic section and the axis is called the vertex of the conic section.
Centre: The point which bisects every chord of the conic passing through it, is called the centre of the conic section.
Latus-rectum: The latus rectum of a conic is the chord passing through the focus and perpendicular to the axis.
Focal Chord: Any chord passing through the focus is called a focal chord of the conic section.
Double Ordinate: A chord perpendicular to the axis of a conic is known as its double ordinate.
The standard equation of parabola is
y2 = 4ax
(a, 0) is the focus of the parabola.
An ellipse is the focus of a point in a plane which moves in the plane in such a way that the ratio of its distance from a fixed point (called focus) in the same plane to its distance from a fixed straight line (called directrix) is always constant which is always less than unity.
The equation of ellipse whose focus is (ae, 0) is
b2 = a2 (1 - e2)
Vertices: The point A and A', where the curve meets the line joining the foci S and S', are called the vertices of the ellipse. The coordinates of A and A' are (a, 0) and (–a, 0) respectively.
Major and Minor Axes
The distance AA' = 2a and BB' = 2b are called the major and minor axes of the ellipse respectively. Since, e < 1 and b2 = a2(1-e2). Therefore, a > b.
The points S(ae, 0) and S'(–ae, 0) are the foci of the ellipse.
ZK and Z'K' are two directrices of the ellipse and their equations are x = a/e and x = - a/e respectively.
A hyperbola is the locus of a point in a plane which moves in the plane in such a way that the ratio of its distance from a fixed point (called focus) in the same plane to its distance from fixed line (called directrix) is always constant which is always greater than unity.
Standard equation of hyperbola is:
b2 = a2 (e2 - 1)