# Coordinate Geometry

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.

### 1. Straight Lines

- Equation of Straight Line
- Parametric Form of Straight Line
- Perpendicular Distance From Point to Straight Line
- Angle Between Two Straight Lines
- Condition for Three Lines To Be Concurrent
- Pair of Straight Lines

### 2. Circle

- Equation of a Circle
- General Equation of Circle
- Parametric Form of Circle
- Equation of Tangent to a Circle
- Length of Tangent to Circle
- Condition For Line To Be Tangent To Circle
- Equation of Chord of Contact of Tangents
- Circles Touching Each Other
- Orthogonal Circles

### 3. Conic Sections

**Parabola**

**Ellipse**

**Hyperbola**

### Cartesian Plane

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.

**Distance Formula**

The distance between P(x_{1}, y_{1}) and Q(x_{2}, y_{2}) is

**Mid-Point Formula**

The mid-point of the line segment joining the points P(x_{1}, y_{1}) and Q(x_{2}, y_{2}) is

**Area of Triangle**

If A(x_{1}, y_{1}), B(x_{2} , y_{2}) and C(x_{3}, y_{3}) be three vertices of a triangle ABC, then its area is given by

**Colinear Points**

Three points A, B and C are said to be collinear (lying on the same straight line) if:

- AB + BC = AC or AC + CB = AB or AB + AC = BC
- The area of the triangle formed by A, B and C is zero

**Section Formula**

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

### Straight Lines

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:

- 2x + 3y + 6 = 0
- 3y + 5 = 0 [a = 0]. The line is parallel to x-axis
- 2x + 7 = 0 [b = 0]. The lines is parallel to y-axis
- 2x + 3y = 0 [c = 0]. The line passes through the origin

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° = ∞.

**Parallel Lines**

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.

**Perpendicular Lines**

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 m_{1} and m_{2} be θ, then

tan θ = (m_{1} - m_{2})/(1 + m_{1}m_{2})

### Standard Forms of the Equation of a Line

**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 (x_{1}, y_{1}) and (x_{2}, y_{2}) is

**Point Slope Form**

The equation of a straight line passing through the point (x_{1}, y_{1}) and having slope m is

(y – y_{1}) = m(x – x_{1}).

**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.

### Circles

The equation of circle whose centre is (h, k) and radius r is given by

(x - h)^{2} + (y - k)^{2} = r^{2}

The equation of circle whose centre is origin and radius r is

x^{2} + y^{2} = r^{2}

**General Equation**

The equation x^{2} + y^{2} + 2gx + 2fy + c = 0 represents a circle of radius of √(g^{2} + f^{2} - c)

- If g
^{2}+ f^{2}- c > 0, then the radius of the circle is real and hence the circle is also real. - If g
^{2}+ f^{2}- c = 0, then the radius of the circle is zero. Such a circle is known a point circle. - If g
^{2}+ f^{2}- c < 0, then the radius of circle is imaginary and such a circle is called an imaginary circle.

### Conic Sections

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.

- If e = 1, the conic is called a parabola
- If e < 1, the conic is called an ellipse
- If e > 1, the conic is called a hyperbola

**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 ax^{2} + 2hxy + by^{2} + 2gx + 2fy + c = 0 will represent

- circle, if a = b and h = 0
- parabola, if h
^{2}= ab - ellipse, if h
^{2}< ab - hyperbola, if h
^{2}> ab - rectangular hyperbola, if h
^{2}> ab and a + b = 0

**Useful Terms**

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.

### Parabola

The standard equation of parabola is

y^{2} = 4ax

(a, 0) is the focus of the parabola.

**Parabola Types**

### Ellipse

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

b^{2} = a^{2} (1 - e^{2})

**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 b^{2} = a^{2}(1-e^{2}). Therefore, a > b.

**Foci**

The points S(ae, 0) and S'(–ae, 0) are the foci of the ellipse.

**Directrices**

ZK and Z'K' are two directrices of the ellipse and their equations are x = a/e and x = - a/e respectively.

### Hyperbola

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:

b^{2} = a^{2} (e^{2} - 1)