Analytical Geometry

Analytical Geometry is a combination of algebra and geometry. In analytical geometry, we aim at presenting the geometric figures using algebraic equations in a two-dimensional coordinate system or in a three-dimensional space. Analytical geometry includes the basic formulas of coordinate geometry, equations of a line and curves, translation and rotation of axes, and three-dimensional geometry concepts.

Let us understand the various sub-branches of analytical geometry, and also check the examples and faqs on analytical geometry.

What Is Analytical Geometry?

Analytical geometry is an important branch of math, which helps in presenting the geometric figures in a two-dimensional plane and to learn the properties of these figures. Here we shall try to know about the coordinate plane and the coordinates of a point, to gain an initial understanding of Analytical geometry.

Coordinate Plane

A cartesian plane divides the plane space into two dimensions and is useful to easily locate the points. It is also referred to as the coordinate plane. The two axes of the coordinate plane are the horizontal x-axis and the vertical y-axis. These coordinate axes divide the plane into four quadrants, and the point of intersection of these axes is the origin (0, 0). Further, any point in the coordinate plane is referred to by a point (x, y), where the x value is the position of the point with reference to the x-axis, and the y value is the position of the point with reference to the y-axis.

The properties of the point represented in the four quadrants of the coordinate plane are:

• The origin O is the point of intersection of the x-axis and the y-axis and has the coordinates (0, 0).
• The x-axis to the right of the origin O is the positive x-axis and to the left of the origin, O is the negative x-axis. Also, the y-axis above the origin O is the positive y-axis, and below the origin O is the negative y-axis.
• The point represented in the first quadrant (x, y) has both positive values and is plotted with reference to the positive x-axis and the positive y-axis.
• The point represented in the second quadrant is (-x, y) is plotted with reference to the negative x-axis and positive y-axis.
• The point represented in the third quadrant (-x, -y) is plotted with reference to the negative x-axis and negative y-axis.
• The point represented in the fourth quadrant (x, -y) is plotted with reference to the positive x-axis and negative y-axis.

Coordinates of a Point

A coordinates is an address, which helps to locate a point in space. For a two-dimensional space, the coordinates of a point are (x, y). Here let us take note of these two important terms.

• Abscissa: It is the x value in the point (x, y), and it is the distance of this point along the x-axis, from the origin
• Ordinate: It is the y value in the point (x, y)., and it is the perpendicular distance of the point from the x-axis, which is parallel to the y-axis.

The coordinates of a point are useful to perform numerous operations of finding distance, midpoint, the slope of a line, equation of a line.

Analytical Geometry - Translation and Rotation of Axes

The coordinate axes in analytical geometry can be translated by moving the axes such that the new axes are parallel to the old axes. Also the coordinates axes can also be rotated at an angle about the origin, with respect to the x-axis. Let us know more about the translation and rotation of axes in the below sentences.

Translation of Axes

The given coordinate axes with the origin as O has the coordinates of a point as (x, y). Here we transfer the origin to a new origin O' located at the point (h, k) with respect to the old coordinate axes. The new coordinate axes is translated such that the new axes are parallel to the old axes. The coordinates of a point transforms from (x, y) to (x' + h, y' + k). Any equation of a line or a curve with respect to the old axes, can be easily changed with reference to the new axes by simply replacing (x, y), in the equation with (x + h, y + k).

Rotation of Axes

The coordinate axes ox and oy are rotated by an angle θ in the anti-clockwise direction, to obtain the new axes ox' and oy'. This coordinates of a point with reference to the old axes is (x, y), and on rotation, the coordinates with reference to the new axes is (x', y'). Further, we can get back the old coordinates by replacing (x', y') as (xCosθy -Sinθ, xSinθ + ycosθ).

Analytical Geometry Formulas in a Coordinate Plane

The formulas of coordinate geometry help in conveniently proving the various properties of lines and figures represented in the coordinate axes. The important formulas of coordinate geometry are the distance formula, slope formula, midpoint formula, and section formula. Let us know more about each of the formulas in the below paragraphs.

Distance Formula

The distance between two points $(x_1,y_1)$ and $x_2,y_2)$ is equal to the square root of the sum of the squares of the difference of the x coordinates and difference of the y-coordinates of the two given points. The formula to find the distance between two given points is as follows.

D = $\sqrt{(x_2-x_1{)}^2+(y_2-y_1{)}^2}$

Slope Formula

The slope of a line is the inclination of the line. The slope can be calculated from the angle made by the line with the positive x-axis, or by taking any two points on the line. The slope of a line inclined at an angle θ with the positive x-axis is m = Tanθ. The slope of a line joining the two points $(x_1,y_1)$ and $x_2,y_2)$ is equal to m = $\frac{(y_2-y_1)}{(x_2-x_1)}$.

Mid-Point Formula

The formula to find the midpoint of the line joining the points $(x_1,y_1)$ and $x_2,y_2)$ is a new point, whose abscissa is the average of the x values of the two given points, and the ordinate is the average of the y values of the two given points. The midpoint lies on the line joining the two points and is located exactly between the two points.

$(x,y)=(\frac{x_1+x_2}{2},\frac{y_1+y_2}{2})$

Section Formula in Coordinate Geometry

The section formula is useful to find the coordinates of a point that divides the line segment joining the points $(x_1,y_1)$ and $(x_2,y_2)$ in the ratio $m:n$. The point dividing the given two points lies on the line joining the two points and is available either between the two points or beyond these two points. The expression for the section formula for the given two points, and the ratio is as follows.

$(x,y)=(\frac{m{x}_2+n{x}_1}{m+n},\frac{m{y}_2+n{y}_1}{m+n})$

Analytical Geometry - Equations of A Line

A set of points in a coordinate plane represents a line. In analytical geometry, the equation of a line helps define all these set of points. There are about five basic different forms of creating an equation of the line. The different forms of the equation of a line are as follows.

• Point Slope Form
• Two Point Form
• Slope-intercept form
• Intercept form
• Normal form

Let us try and understand more about each one of these forms of the equation of a line.

Point-Slope Form

The point-slope form of the equation of a line requires a point on the line and the slope of the line. The referred point on the line is (x1, y1) and the slope of the line is m. The point is a numeric value and representing the x coordinate and the y coordinate of the point and the slope of the line m is the inclination of the line with the positive x-axis. The point-slope form of the equation of a line is (y - y1) = m(x - x1).

Two Point Form

The two-point form of the equation of a line is a further explanation of the point-slope form of equation of a line. In the point-slope form of the equation of a line the slope m = (y2 - y1)/(x2 - x1) is substituted to form the two-point form of the equation of a line. The equation of a line passing through the two points (x1, y1), and (x2, y2) is as follows.

$$(y-y_1)=\frac{(y_2-y_1)}{(x_2-x_1)}(x-x_1)$$

Slope Intercept Form

The slope-intercept form of a line is y = mx + c. Here m is the slope of the line and 'c' is the y-intercept of the line. This line cuts the y-axis at the point (0, c) and c is the distance of this point on the y-axis from the origin. The slope-intercept form of the equation of a line is an important form and has great applications in different topics of mathematics and engineering.
y = mx + c

Intercept Form

The equation of a line in intercept form is formed with the x-intercept 'a' and the y-intercept 'b'. The line cuts the x-axis at the point (a, 0), and the y-axis at the point(0, b), and a, b are the respective distances of these points from the origin. Further, these two points can be substituted in the two-point form of the equation of a line and simplified to get this intercept form of the equation of the line. This intercept form explains the distance at which the line cuts the x-axis and the y-axis from the origin.
$\frac{x}{a}+\frac{y}{b}=1$

Normal Form

The normal form of the equation of a line is based on the perpendicular to the line, which passes through the origin. The line perpendicular to the given line, and which passes through the origin is called the normal. Here the length of the normal is 'p' and the angle made by this normal with the positive x-axis is 'θ'. The equation of the normal form of the equation of a line is xcosθ + ysinθ = p.

Analytical Geometry - Conic Section

The conic section in analytical geometry represents the curves that have been formed from curved lines, and have been defined with reference to a fixed point called the focus and the fixed-line called the directrix. The important conics are the circle, parabola, ellipse and the hyperbola. The standard form of equations of the different conics is as follows.

• Circle: x²+y²= a²
• Parabola: y²= 4ax when a>0
• Ellipse: x²/a² + y²/b² = 1
• Hyperbola: x²/a² – y²/b² = 1

Circle

The circle has a center and radius. A circle represents the locus of a point such that it's distance from a fixed point called the center is equal to a constant value called the radius. The general equation of a circle having the center at (h, k), and having a radius of r units is (x - h)² + (y - k)² = r². Further, the standard equation of a circle having the center at (0, 0), and the radius of 'a' units is x²+y²= a².

Parabola

A parabola refers to an equation of a curve, such that a point on the curve is equidistant from a fixed point, and a fixed-line. The fixed point is called the focus of the parabola, and the fixed line is called the directrix of the parabola. A locus of any point which is equidistant from a given point (focus) and a given line (directrix) is called a parabola. The general equation of a parabola is: y = a(x-h)² + k or x = a(y-k)² +h, where (h,k) denotes the vertex. The standard equation of a regular parabola is y² = 4ax.

Ellipse

An ellipse in math is the locus of a plane point in such that its distance from a fixed point has a constant ratio 'e' to its distance from a fixed line, which is less than 1. The fixed point is called the focus and is denoted by S, the constant ratio $e$ is the eccentricity, and the fixed line is called as directrix (d) of the ellipse. Also, an ellipse is the locus of a point, the sum of whose distances from two fixed points is a constant value. The two fixed points are called the foci of the ellipse. The standard equation of an ellipse is $\frac{x^2}{a^2}+\frac{y^2}{b^2}=1$

Hyperbola

A hyperbola is a set of points whose difference of distances from two foci is a constant value. This difference is taken from the distance from the farther focus and then the distance from the nearer focus. For a point P(x, y). on the hyperbola and for two foci F, F', the locus of the hyperbola is PF - PF' = 2a. The equation $\frac{x^2}{a^2}-\frac{y^2}{b^2}=1$ represents the standard form of the equation of a hyperbola. Here the x-axis is the transverse axis of the hyperbola, and the y-axis is the conjugate axis of the hyperbola.

Analytical Geometry in Three Dimensional Space

The space around us can be visualized as a three-dimensional space with the help of the x-axis, y-axis, and z-axis respectively. This is useful to present the equations of a line and a plane respectively.

Direction Ratios & Direction Cosines

The line passing through the origin and passing through the point (a, b, c) has direction ratios of a, b, c respectively. Further, this line makes an angle α, β, γ with reference to the x-axis, y-axis, y-axis, then Cosα, Cosβ, Cosγ are called the direction cosines of the line.

These direction cosines are represented by l, m, n, and we have $l=\pm \frac{a}{\sqrt{a^2+b^2+c^2}}$, $m=\pm \frac{b}{\sqrt{a^2+b^2+c^2}}$, $n=\pm \frac{c}{\sqrt{a^2+b^2+c^2}}$.

Equation of a Line

The equation of a line can be calculated in two ways using the following formulas.

1. The equation of a line passing through a given point $\overrightarrow{a}$ and parallel to the vector $\overrightarrow{b}$ is $\overrightarrow{r}.=\overrightarrow{a}+\lambda \overrightarrow{b}$.
2. The equation of a line passing through two given points $\overrightarrow{a}$, and $\overrightarrow{b}$ is $\overrightarrow{r}.=\overrightarrow{a}+\lambda (\overrightarrow{b}-\overrightarrow{a})$.

Equation of a Plane

There are four different ways of writing the equation of a plane, based on the given input values.

1. The equation of plane at a distance of d units from the origin, and having a normal n is $\overrightarrow{r}.\hat{n}=d$.
2. The equation of a plane passing through a point $\overrightarrow{a}$, and having a normal vector $\overrightarrow{N}$ is $(\overrightarrow{r}-\overrightarrow{a}).\overrightarrow{N}=0$.
3. The equation of a plane passing through three vector points $\overrightarrow{a}$, $\overrightarrow{b}$, and $\overrightarrow{c}$ is $(\overrightarrow{r}-\overrightarrow{a}).\left[\right(\overrightarrow{b}-\overrightarrow{a})\times (\overrightarrow{c}-\overrightarrow{a}\left)\right]=0$.
4. The equation of a plane passing through two planes having the normal vectors as ${\overrightarrow{n}}_1$, ${\overrightarrow{n}}_2$ and distances from the origin as $d_1$, $d_2$ respectively is $\overrightarrow{r}.({\overrightarrow{n}}_1+\lambda {\overrightarrow{n}}_2)=d_1+\lambda d_2$.

Angle Between Two Line and Two Planes

The angle between two lines and two planes can be calculate using the following set of formulas.

1. The angle between two lines having direction ratios $a_1,b_1,c_1$, and $a_2,b_2,c_2$ respectively is $Cos\theta =\left|\frac{a_1.a_2+b_1.b_2+c_1.c_2}{\sqrt{a_1^2+b_1^2+c_1^2}.\sqrt{a_2^2+b_2^2+c_2^2}}\right|$.
2. The angle between two planes $A_{1}x+B_{1}y+C_{1}Z+D_1=0$, $A_{2}x+B_{2}y+C_{2}Z+D_2=0$ is $Cos\theta =\left|\frac{A_1.A_2+B_1.B_2+C_1.C_2}{\sqrt{A_1^2+B_1^2+C_1^2}.\sqrt{A_2^2+B_2^2+C_2^2}}\right|$.

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