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 twodimensional coordinate system or in a threedimensional space. Analytical geometry includes the basic formulas of coordinate geometry, equations of a line and curves, translation and rotation of axes, and threedimensional geometry concepts.
Let us understand the various subbranches 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 twodimensional 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 xaxis and the vertical yaxis. 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 xaxis, and the y value is the position of the point with reference to the yaxis.
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 xaxis and the yaxis and has the coordinates (0, 0).
 The xaxis to the right of the origin O is the positive xaxis and to the left of the origin, O is the negative xaxis. Also, the yaxis above the origin O is the positive yaxis, and below the origin O is the negative yaxis.
 The point represented in the first quadrant (x, y) has both positive values and is plotted with reference to the positive xaxis and the positive yaxis.
 The point represented in the second quadrant is (x, y) is plotted with reference to the negative xaxis and positive yaxis.
 The point represented in the third quadrant (x, y) is plotted with reference to the negative xaxis and negative yaxis.
 The point represented in the fourth quadrant (x, y) is plotted with reference to the positive xaxis and negative yaxis.
Coordinates of a Point
A coordinates is an address, which helps to locate a point in space. For a twodimensional 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 xaxis, 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 xaxis, which is parallel to the yaxis.
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 xaxis. 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 anticlockwise 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 ycoordinates 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 xaxis, or by taking any two points on the line. The slope of a line inclined at an angle θ with the positive xaxis 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 = \( \frac {(y_2  y_1)}{(x_2  x_1)} \).
MidPoint 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.
\((x, y) =\left(\dfrac{x_1 + x_2}{2}, \dfrac{y_1 + y_2}{2}\right)\)
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.
\((x, y) = \left(\frac{mx_2 + nx_1}{m + n}, \frac{my_2 + ny_1}{m + n}\right) \)
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
 Slopeintercept form
 Intercept form
 Normal form
Let us try and understand more about each one of these forms of the equation of a line.
PointSlope Form
The pointslope 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 (x_{1}, y_{1}) 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 xaxis. The pointslope form of the equation of a line is (y  y_{1}) = m(x  x_{1}).
Two Point Form
The twopoint form of the equation of a line is a further explanation of the pointslope form of equation of a line. In the pointslope form of the equation of a line the slope m = (y_{2}  y_{1})/(x_{2}  x_{1}) is substituted to form the twopoint form of the equation of a line. The equation of a line passing through the two points (x_{1}, y_{1}), and (x_{2}, y_{2}) is as follows.
\[(y y_1) = \frac{(y_2  y_1)}{(x_2  x_1)}(x  x_1) \]
Slope Intercept Form
The slopeintercept form of a line is y = mx + c. Here m is the slope of the line and 'c' is the yintercept of the line. This line cuts the yaxis at the point (0, c) and c is the distance of this point on the yaxis from the origin. The slopeintercept 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 xintercept 'a' and the yintercept 'b'. The line cuts the xaxis at the point (a, 0), and the yaxis 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 twopoint 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 xaxis and the yaxis 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 xaxis 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 fixedline 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: x^{2}+y^{2}= a^{2}
 Parabola: y^{2}= 4ax when a>0
 Ellipse: x^{2}/a^{2} + y^{2}/b^{2} = 1
 Hyperbola: x^{2}/a^{2} – y^{2}/b^{2} = 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} = r^{2}. Further, the standard equation of a circle having the center at (0, 0), and the radius of 'a' units is x^{2}+y^{2}= a^{2}.
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 fixedline. 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(xh)^{2} + k or x = a(yk)^{2} +h, where (h,k) denotes the vertex. The standard equation of a regular parabola is y^{2} = 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 \(\dfrac{x^2}{a^2} + \dfrac{y^2}{b^2} = 1\)
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 \(\dfrac{x^2}{a^2}  \dfrac{y^2}{b^2} = 1\) represents the standard form of the equation of a hyperbola. Here the xaxis is the transverse axis of the hyperbola, and the yaxis is the conjugate axis of the hyperbola.
Analytical Geometry in Three Dimensional Space
The space around us can be visualized as a threedimensional space with the help of the xaxis, yaxis, and zaxis 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 xaxis, yaxis, yaxis, then Cosα, Cosβ, Cosγ are called the direction cosines of the line.
These direction cosines are represented by l, m, n, and we have \(l =\pm \dfrac{a}{\sqrt{a^2 + b^2 + c^2}}\), \(m =\pm \dfrac{b}{\sqrt{a^2 + b^2 + c^2}}\), \(n =\pm \dfrac{c}{\sqrt{a^2 + b^2 + c^2}}\).
Equation of a Line
The equation of a line can be calculated in two ways using the following formulas.
 The equation of a line passing through a given point \(\overrightarrow a\) and parallel to the vector \(\overrightarrow b\) is \(\overrightarrow r. =\overrightarrow a +λ\overrightarrow b\).
 The equation of a line passing through two given points \(\overrightarrow a\), and \(\overrightarrow b\) is \(\overrightarrow r. =\overrightarrow a +λ(\overrightarrow b  \overrightarrow a)\).
Equation of a Plane
There are four different ways of writing the equation of a plane, based on the given input values.
 The equation of plane at a distance of d units from the origin, and having a normal n is \(\overrightarrow r.\hat n = d\).
 The equation of a plane passing through a point \(\overrightarrow a\), and having a normal vector \(\overrightarrow N\) is \((\overrightarrow r  \overrightarrow a).\overrightarrow N = 0\).
 The equation of a plane passing through three vector points \(\overrightarrow a\), \(\overrightarrow b\), and \(\overrightarrow c\) is \((\overrightarrow r  \overrightarrow a).[(\overrightarrow b  \overrightarrow a) \times (\overrightarrow c  \overrightarrow a)] = 0\).
 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 \(\overrightarrow r. (\overrightarrow n_1 +λ\overrightarrow n_2 ) = d_1 + λd_2\).
Angle Between Two Line and Two Planes
The angle between two lines and two planes can be calculate using the following set of formulas.
 The angle between two lines having direction ratios \(a_1, b_1, c_1\), and \(a_2, b_2, c_2\) respectively is \(Cosθ =\left \dfrac{a_1.a_2 +b_1.b_2+c_1.c_2}{\sqrt{a_1^2 + b_1^2+c_1^2}.\sqrt{a_2^2 + b_2^2+c_2^2}}\right \).
 The angle between two planes \(A_1x +B_1y+C_1Z + D_1 = 0\), \(A_2x +B_2y+C_2Z + D_2 = 0\) is \(Cosθ =\left \dfrac{A_1.A_2 +B_1.B_2+C_1.C_2}{\sqrt{A_1^2 + B_1^2+C_1^2}.\sqrt{A_2^2 + B_2^2+C_2^2}}\right \).
Related Topics
The following topics help in more clearly understanding the concepts of analytical geometry.
Examples on Analytical Geometry

Example 1: Find the equation of a line in analytical geometry, having the xintercept of 5 units, and yintercept of 6 units respectively.
Solution:
The given intercepts of the xaxis is a = 5, and of yaxis is b = 6.
The required equation of the straight line is x/a + y/b = 1.
x/5 + y/6 = 1
6x + 5y = 30
Therefore the required equation of the line is 6x + 5y = 30.

Example 2: Find the coordinates of the midpoint of the line joining the points (4, 3, 2), and (2, 1, 5). Use the midpoint formula of analytical geometry in threedimensional space.
Solution:
The given points are (4, 3, 2), and (2, 1, 5).
Using the midpoint formula we have MP = \(\left(\frac{x_1 + x_2}{2}, \frac{y_1 + y_2}{2}, \frac{z_1 + z_2}{2} \right)\)
MP = \(\left(\frac{4 + 2}{2}, \frac{(3) + 1}{2}, \frac{2 + 5}{2} \right)\)
MP = (3, 1, 7/2).
Therefore the coordinates of the midpoint are (3, 1, 7/2).
Practice Questions on Analytical Geometry
Here are a few activities for you to practice. Select your answer and click the "Check Answer" button to see the result.
FAQs on Analytical Geometry
What Is 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 twodimensional coordinate system or in a threedimensional space.
What Are the Topics in Analytical Geometry?
The topics of analytical geometry include coordinate geometry, threedimensional geometry, vectors. Here it also includes topics of translation and rotation of axes, equation of line and equation of curves, equation of a line and plane in threedimensional geometry.
What Is the Fundamental Principle of Analytical Geometry?
The fundamental principle of analytical geometry is based on the principle of geometry and algebra. In analytical geometry, we use the distance formula, midpoint formula, section formula, slope formula, in a coordinate plane, and in a threedimensional plane.
What Is the Difference Between Analytical Geometry And Plane Geometry?
Analytical geometry uses the concepts of geometry and algebra and represents the lines, curves, conics as algebraic expressions. Geometry is the study of the shapes and properties of geometric figures. Geometry form the foundation for analytical geometry.
How Do you Do Analytical Geometry?
The analytical geometry is solved using algebraic concepts of solving equations. Here we use the basic distance formula, midpoint formula, section formula, equation of line, and curve formula to represent the geometric figures, which are further solved using algebraic concepts.
How Is Analytical Geometry Different From Coordinate Geometry?
Coordinate geometry is a subtopic of analytical geometry. Analytical geometry also includes topics of threedimensional geometry and vectors.
Is Three Dimensional Geometry Part of Analytical Geometry?
The threedimensional geometry is a part of analytical geometry. The lines or planes in threedimensional space are represented using the concepts of analytical geometry.
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