NDA & Naval Academy - Analytical Geometry of two and three dimensions

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NDA & Naval Academy Elementary Mathematics

Analytical Geometry of two and three dimensions

Analytical Geometry, also called Coordinate Geometry, connects algebra and geometry. It uses numbers, coordinates, equations, and graphs to study points, lines, circles, conics, planes, spheres, and three-dimensional objects.

Elementary Mathematics Analytical Geometry 2D and 3D Geometry Competitive Exams

Analytical Geometry, also called Coordinate Geometry, connects algebra and geometry. It uses numbers, coordinates, equations, and graphs to study points, lines, circles, conics, planes, spheres, and three-dimensional objects. This chapter is highly useful for competitive examinations because many geometry problems can be solved faster using formulas and coordinate methods.

What is Analytical Geometry?

Analytical Geometry represents geometrical figures using algebraic equations. A point is represented by coordinates, a line by a linear equation, a circle by a quadratic equation, and three-dimensional objects by equations involving \(x\), \(y\), and \(z\).

In two dimensions, we mainly use the rectangular Cartesian coordinate system with two axes: the x-axis and the y-axis. In three dimensions, we use three mutually perpendicular axes: x-axis, y-axis, and z-axis.

Quick idea: Analytical Geometry converts diagrams into equations. Once a figure is represented algebraically, we can calculate distances, angles, slopes, intersections, and positions accurately.

Area	What It Covers	Exam Focus
2D Coordinate Geometry	Points, distance formula, section formula, slope, line equations.	Distance, midpoint, slope, equation of line.
Straight Lines	Different forms of line equation and angle between lines.	Slope form, intercept form, normal form, distance from point to line.
Circle	Standard and general equation of a circle.	Centre, radius, equation formation.
Conic Sections	Parabola, ellipse, hyperbola, eccentricity, axes.	Standard forms and basic properties.
3D Geometry	Points in space, direction cosines, direction ratios, lines, planes, spheres.	Distance, angle, equation of line, plane, sphere.

“Coordinate geometry makes geometry measurable, visual, and algebraically solvable.”

Mathematics Tip

Key points

Understand Cartesian coordinates.
Memorize distance and midpoint formulas.
Learn different forms of line equations.
Practice angle between two lines.
Know standard equation of circle and conics.
Understand direction cosines and direction ratios.
Learn equations of lines, planes, and spheres in 3D.

coordinates lines circle conics planes sphere

Rectangular Cartesian Coordinate System

The rectangular Cartesian coordinate system is used to locate points in a plane. It consists of two perpendicular lines called axes. The horizontal axis is the x-axis and the vertical axis is the y-axis. Their point of intersection is called the origin.

Origin

The point where the x-axis and y-axis meet. It is represented as \(O(0,0)\).

Coordinates

A point in a plane is written as \(P(x,y)\), where \(x\) is abscissa and \(y\) is ordinate.

Quadrants

The coordinate plane is divided into four regions called quadrants.

Signs

Quadrants have signs: I \((+,+)\), II \((-,+)\), III \((-,-)\), IV \((+,-)\).

Rule: Always write coordinates in the order \((x,y)\) in two dimensions and \((x,y,z)\) in three dimensions.

Formula Bank: Two Dimensions

Distance Formula

Between \(A(x_1,y_1)\) and \(B(x_2,y_2)\):
\(AB=\sqrt{(x_2-x_1)^2+(y_2-y_1)^2}\)

Midpoint Formula

Midpoint of \(A(x_1,y_1)\), \(B(x_2,y_2)\):
\(\left(\frac{x_1+x_2}{2},\frac{y_1+y_2}{2}\right)\)

Slope of a Line

\(m=\frac{y_2-y_1}{x_2-x_1}\), where \(x_2 \neq x_1\)

Angle Between Lines

If slopes are \(m_1\) and \(m_2\):
\(\tan \theta=\left|\frac{m_2-m_1}{1+m_1m_2}\right|\)

Point-Line Distance

From \((x_1,y_1)\) to \(Ax+By+C=0\):
\(d=\frac{|Ax_1+By_1+C|}{\sqrt{A^2+B^2}}\)

Circle

Centre \((h,k)\), radius \(r\):
\((x-h)^2+(y-k)^2=r^2\)

Tip: In 2D coordinate geometry, most questions are solved using distance, slope, and equation of line.

Analytical Geometry Formula Selection Guide

Analytical Geometry questions become easier when the student first identifies the object involved: point, line, circle, conic, plane or sphere. The table below helps select the correct formula quickly.

Question Type	What to Use	Typical Clue
Distance between two points	Distance formula	Two points are given
Middle point of a segment	Midpoint formula	Find midpoint or centre of a segment
Direction of a line	Slope formula	Two points or line equation is given
Equation of a line	Point-slope, two-point, intercept or general form	Point, slope, intercepts or two points are given
Distance from point to line	\(d=\frac{\|Ax_1+By_1+C\|}{\sqrt{A^2+B^2}}\)	Point and line \(Ax+By+C=0\) are given
Circle question	Standard or general equation of circle	Centre, radius, or equation involving \(x^2+y^2\)
Conic question	Parabola, ellipse or hyperbola standard form	Eccentricity, focus, directrix, axes or standard equation
Three-dimensional distance	3D distance formula	Points are given as \((x,y,z)\)
Line in three dimensions	Direction ratios and symmetric form	Two 3D points or point with direction ratios
Plane in three dimensions	\(Ax+By+Cz+D=0\)	Plane equation or normal direction ratios
Sphere question	Standard or general equation of sphere	Equation involving \(x^2+y^2+z^2\)

Exam shortcut: First decide whether the question is in 2D or 3D. Then identify the object and apply the matching formula. Most mistakes happen due to wrong sign substitution or using a 2D formula in a 3D problem.

Equation of a Line in Various Forms

A straight line in a plane can be represented in different forms depending on the information given in the question. Choosing the right form saves time.

Form	Equation	When to Use
Slope-Intercept Form	\(y=mx+c\)	When slope and y-intercept are known.
Point-Slope Form	\(y-y_1=m(x-x_1)\)	When one point and slope are known.
Two-Point Form	\(y-y_1=\frac{y_2-y_1}{x_2-x_1}(x-x_1)\)	When two points are given.
Intercept Form	\(\frac{x}{a}+\frac{y}{b}=1\)	When x-intercept and y-intercept are known.
General Form	\(Ax+By+C=0\)	Most common general representation of a line.
Normal Form	\(x\cos \alpha+y\sin \alpha=p\)	When perpendicular distance from origin and angle are used.

Important: If two lines have slopes \(m_1\) and \(m_2\), then they are parallel when \(m_1=m_2\), and perpendicular when \(m_1m_2=-1\).

Circle: Standard and General Form

A circle is the locus of a point that moves in a plane such that its distance from a fixed point remains constant. The fixed point is called the centre and the constant distance is called the radius.

Type	Equation	Meaning
Standard Form	\((x-h)^2+(y-k)^2=r^2\)	Centre is \((h,k)\), radius is \(r\).
Centre at Origin	\(x^2+y^2=r^2\)	Centre is \((0,0)\), radius is \(r\).
General Form	\(x^2+y^2+2gx+2fy+c=0\)	Centre is \((-g,-f)\).
Radius in General Form	\(r=\sqrt{g^2+f^2-c}\)	Used after identifying \(g\), \(f\), and \(c\).

Exam tip: To convert general form into standard form, complete the square separately for \(x\) and \(y\).

Conic Sections: Parabola, Ellipse and Hyperbola

A conic section is a curve obtained by cutting a cone with a plane. The important conics are parabola, ellipse, and hyperbola. Each conic has a focus, directrix, axis, and eccentricity.

Conic	Standard Form	Eccentricity	Basic Nature
Parabola	\(y^2=4ax\) or \(x^2=4ay\)	\(e=1\)	Open curve with one focus and one directrix.
Ellipse	\(\frac{x^2}{a^2}+\frac{y^2}{b^2}=1\), where \(a>b\)	\(0<e<1\)</td>	Closed oval curve with two foci.
Hyperbola	\(\frac{x^2}{a^2}-\frac{y^2}{b^2}=1\)	\(e>1\)	Open curve with two separate branches.
Rectangular Hyperbola	\(xy=c^2\)	Special case	Asymptotes are perpendicular.

Axis of a Conic

The axis is the line of symmetry of a conic. In standard forms, it is usually the x-axis or y-axis depending on the orientation of the conic.

Eccentricity

Eccentricity measures how much a conic deviates from circular shape. It is denoted by \(e\). For circle \(e=0\), ellipse \(0<e<1\), parabola \(e=1\), and hyperbola \(e>1\).

Remember: Parabola has eccentricity 1, ellipse has eccentricity less than 1, and hyperbola has eccentricity greater than 1.

Formula Bank: Three Dimensions

In three-dimensional geometry, a point is represented as \(P(x,y,z)\). The three coordinates show the position of the point with respect to the x-axis, y-axis, and z-axis.

Concept	Formula	Use
Distance Between Two Points	\(AB=\sqrt{(x_2-x_1)^2+(y_2-y_1)^2+(z_2-z_1)^2}\)	Distance between \(A(x_1,y_1,z_1)\) and \(B(x_2,y_2,z_2)\).
Direction Ratios	\(a,b,c\)	Numbers proportional to the direction of a line.
Direction Cosines	\(l=\cos \alpha,\ m=\cos \beta,\ n=\cos \gamma\)	Cosines of angles made by a line with positive coordinate axes.
Relation	\(l^2+m^2+n^2=1\)	Important relation among direction cosines.
D.C. from D.R.	\(l=\frac{a}{\sqrt{a^2+b^2+c^2}},\ m=\frac{b}{\sqrt{a^2+b^2+c^2}},\ n=\frac{c}{\sqrt{a^2+b^2+c^2}}\)	Convert direction ratios into direction cosines.
Sphere	\((x-a)^2+(y-b)^2+(z-c)^2=r^2\)	Sphere with centre \((a,b,c)\) and radius \(r\).

Important: In 3D, distance formula is an extension of 2D distance formula with one extra coordinate \(z\).

Equation of a Line in Three Dimensions

A line in three-dimensional space can be represented in different forms. The equation depends on whether we know a point and direction ratios, or two points on the line.

Form	Equation	Meaning
Symmetric Form	\(\frac{x-x_1}{a}=\frac{y-y_1}{b}=\frac{z-z_1}{c}\)	Line through \((x_1,y_1,z_1)\) with direction ratios \(a,b,c\).
Parametric Form	\(x=x_1+ar,\ y=y_1+br,\ z=z_1+cr\)	\(r\) is a parameter.
Line Through Two Points	\(\frac{x-x_1}{x_2-x_1}=\frac{y-y_1}{y_2-y_1}=\frac{z-z_1}{z_2-z_1}\)	Used when two points are given.

Exam tip: If two points are given, first find direction ratios: \(x_2-x_1,\ y_2-y_1,\ z_2-z_1\).

Equation of a Plane in Various Forms

A plane is a flat surface that extends infinitely in all directions. In 3D geometry, a plane is usually represented by a linear equation in \(x\), \(y\), and \(z\).

Form	Equation	When to Use
General Form	\(Ax+By+Cz+D=0\)	Most common form of a plane.
Normal Form	\(lx+my+nz=p\)	When direction cosines of normal and distance from origin are known.
Intercept Form	\(\frac{x}{a}+\frac{y}{b}+\frac{z}{c}=1\)	When intercepts on coordinate axes are known.
Point-Normal Form	\(A(x-x_1)+B(y-y_1)+C(z-z_1)=0\)	When a point and normal vector are known.

Important: In the plane \(Ax+By+Cz+D=0\), the normal direction ratios are \(A,B,C\).

Angles in Three Dimensions

Angle questions in 3D are commonly based on direction ratios, direction cosines, and normal vectors of planes.

Angle Type	Formula	Use
Angle Between Two Lines	\(\cos \theta=\frac{a_1a_2+b_1b_2+c_1c_2}{\sqrt{a_1^2+b_1^2+c_1^2}\sqrt{a_2^2+b_2^2+c_2^2}}\)	When direction ratios of two lines are known.
Angle Between Two Lines Using D.C.	\(\cos \theta=l_1l_2+m_1m_2+n_1n_2\)	When direction cosines are given.
Angle Between Two Planes	\(\cos \theta=\frac{A_1A_2+B_1B_2+C_1C_2}{\sqrt{A_1^2+B_1^2+C_1^2}\sqrt{A_2^2+B_2^2+C_2^2}}\)	Use coefficients of \(x\), \(y\), and \(z\) as normal direction ratios.

Remember: The angle between two planes is the angle between their normal vectors.

Equation of a Sphere

A sphere is the set of all points in space that are at a fixed distance from a fixed point. The fixed point is called the centre and the fixed distance is called the radius.

Type	Equation	Meaning
Standard Form	\((x-a)^2+(y-b)^2+(z-c)^2=r^2\)	Centre is \((a,b,c)\), radius is \(r\).
Centre at Origin	\(x^2+y^2+z^2=r^2\)	Centre is \((0,0,0)\), radius is \(r\).
General Form	\(x^2+y^2+z^2+2ux+2vy+2wz+d=0\)	Centre is \((-u,-v,-w)\).
Radius in General Form	\(r=\sqrt{u^2+v^2+w^2-d}\)	Used after identifying \(u\), \(v\), \(w\), and \(d\).

Exam tip: A sphere equation in standard form is very similar to the circle equation, but it includes the additional \(z\)-coordinate.

Step-by-Step Solving Method

Step	Action	Example Focus
Step 1	Identify whether the problem is 2D or 3D.	Check whether coordinates are \((x,y)\) or \((x,y,z)\).
Step 2	Identify the object involved.	Point, line, circle, conic, plane, or sphere.
Step 3	Select the correct formula or equation form.	Distance formula, line form, circle form, plane form.
Step 4	Substitute values carefully.	Use coordinates, slopes, direction ratios, or coefficients.
Step 5	Simplify and verify the answer.	Check signs, squares, roots, and units.

Important: Most coordinate geometry mistakes happen because of sign errors and incorrect substitution of coordinates.

Solved Examples

Question	Method	Answer
Find the distance between \(A(1,2)\) and \(B(4,6)\).	\(AB=\sqrt{(4-1)^2+(6-2)^2}\) \(=\sqrt{3^2+4^2}=\sqrt{25}=5\)	5
Find the slope of the line joining \((2,3)\) and \((5,9)\).	\(m=\frac{9-3}{5-2}=\frac{6}{3}=2\)	2
Find the equation of the line with slope 3 and y-intercept 4.	Use \(y=mx+c\). Here \(m=3\), \(c=4\).	\(y=3x+4\)
Find the distance of point \((1,2)\) from the line \(3x+4y-5=0\).	\(d=\frac{\|3(1)+4(2)-5\|}{\sqrt{3^2+4^2}}\) \(=\frac{\|3+8-5\|}{5}=\frac{6}{5}\)	\(\frac{6}{5}\)
Find the centre and radius of \((x-2)^2+(y+3)^2=25\).	Compare with \((x-h)^2+(y-k)^2=r^2\). \(h=2\), \(k=-3\), \(r=5\).	Centre \((2,-3)\), radius 5
Find the distance between \(A(1,2,3)\) and \(B(4,6,3)\).	\(AB=\sqrt{(4-1)^2+(6-2)^2+(3-3)^2}\) \(=\sqrt{9+16+0}=5\)	5
Find direction cosines for direction ratios \(2,3,6\).	Denominator \(=\sqrt{2^2+3^2+6^2}=\sqrt{49}=7\). Direction cosines are \(\frac{2}{7},\frac{3}{7},\frac{6}{7}\).	\(\frac{2}{7}, \frac{3}{7}, \frac{6}{7}\)
Find the centre and radius of \((x-1)^2+(y-2)^2+(z-3)^2=16\).	Compare with \((x-a)^2+(y-b)^2+(z-c)^2=r^2\). Centre \(=(1,2,3)\), radius \(=4\).	Centre \((1,2,3)\), radius 4

Note: Always identify whether the equation represents a line, circle, conic, plane, or sphere before applying formulas.

Common Traps and Shortcuts

Common Traps

Interchanging x-coordinate and y-coordinate.
Using 2D distance formula for 3D points.
Forgetting square root in distance formula.
Making sign errors in point-line distance formula.
Confusing radius and radius squared in circle or sphere equations.
Using slope formula when the line is vertical.
Confusing direction ratios with direction cosines.
Taking angle between planes directly instead of using normals.
Forgetting to complete the square in circle and sphere equations.

Useful Shortcuts

Use distance formula for checking equality of sides.
Use slope to test parallel and perpendicular lines.
Use \(m_1=m_2\) for parallel lines.
Use \(m_1m_2=-1\) for perpendicular lines.
For circle and sphere, compare directly with standard form.
In 3D, direction ratios between two points are coordinate differences.
For planes, coefficients of \(x,y,z\) give normal direction ratios.
For conics, identify standard form before using properties.

Exam approach: Identify whether the problem is based on coordinates, distance, line, circle, conics, direction ratios, plane, or sphere, then apply the correct formula.

Practice

A) Multiple Choice Questions

Find the distance between \((0,0)\) and \((3,4)\).
3 4 5 7
The slope of the line \(y=2x+5\) is:
2 5 -2 \(\frac{1}{2}\)
The centre of the circle \((x-3)^2+(y+2)^2=16\) is:
\((3,2)\) \((-3,2)\) \((3,-2)\) \((-3,-2)\)
The eccentricity of a parabola is:
0 1 Less than 1 Greater than 1
The standard equation of a sphere with centre \((0,0,0)\) and radius \(r\) is:
\(x^2+y^2=r^2\) \(x^2+y^2+z^2=r^2\) \(x+y+z=r\) \(xy+yz+zx=r^2\)

B) Solve the Higher-Order Problems

Find the midpoint of the line segment joining \((2,4)\) and \((6,8)\). (Hint: Use midpoint formula.)
Find the equation of a line passing through \((1,2)\) with slope 3. (Hint: Use point-slope form.)
Find the radius of the circle \(x^2+y^2-4x+6y-12=0\). (Hint: Compare with \(x^2+y^2+2gx+2fy+c=0\).)
Find the distance between \((1,2,3)\) and \((3,4,5)\). (Hint: Use 3D distance formula.)
Find direction cosines corresponding to direction ratios \(1,2,2\). (Hint: Divide each direction ratio by \(\sqrt{1^2+2^2+2^2}\).)

C) Match the Concept with the Correct Rule

Concept	Correct Rule / Meaning
Distance Formula in 2D	\(\sqrt{(x_2-x_1)^2+(y_2-y_1)^2}\)
Slope	\(\frac{y_2-y_1}{x_2-x_1}\)
Circle	Set of points equidistant from a fixed point in a plane
Parabola	Conic with eccentricity 1
Direction Cosines	Cosines of angles made by a line with coordinate axes
Plane	A flat surface represented by \(Ax+By+Cz+D=0\)
Sphere	Set of points equidistant from a fixed point in space
Angle Between Planes	Angle between their normal vectors

Analytical Geometry Reminder

Analytical Geometry helps solve geometry problems using coordinates, equations, formulas, and algebraic methods. In two dimensions, focus on points, lines, circles, and conics. In three dimensions, focus on points in space, direction cosines, direction ratios, lines, planes, angles, and spheres.

Task: Create five analytical geometry questions using one question each from distance formula, equation of line, circle, direction cosines, and equation of sphere.

Show Suggested Answers

Multiple Choice

5
Distance \(=\sqrt{3^2+4^2}=\sqrt{25}=5\).
2
In \(y=mx+c\), slope is \(m\). Therefore, slope is 2.
\((3,-2)\)
Compare with \((x-h)^2+(y-k)^2=r^2\). Centre is \((3,-2)\).
1
Eccentricity of a parabola is 1.
\(x^2+y^2+z^2=r^2\)
This is the equation of a sphere with centre at the origin.

Higher-Order Problems

Midpoint of \((2,4)\) and \((6,8)\):
\(\left(\frac{2+6}{2},\frac{4+8}{2}\right)=(4,6)\).
Answer = \((4,6)\).
Use point-slope form:
\(y-y_1=m(x-x_1)\).
\(y-2=3(x-1)\).
\(y=3x-1\).
Answer = \(y=3x-1\).
Given \(x^2+y^2-4x+6y-12=0\).
Compare with \(x^2+y^2+2gx+2fy+c=0\).
\(2g=-4\), so \(g=-2\).
\(2f=6\), so \(f=3\).
\(c=-12\).
Radius \(=\sqrt{g^2+f^2-c}=\sqrt{4+9+12}=\sqrt{25}=5\).
Answer = 5.
Distance between \((1,2,3)\) and \((3,4,5)\):
\(d=\sqrt{(3-1)^2+(4-2)^2+(5-3)^2}\)
\(=\sqrt{4+4+4}=\sqrt{12}=2\sqrt{3}\).
Answer = \(2\sqrt{3}\).
Direction ratios are \(1,2,2\).
Denominator \(=\sqrt{1^2+2^2+2^2}=\sqrt{9}=3\).
Direction cosines are \(\frac{1}{3},\frac{2}{3},\frac{2}{3}\).
Answer = \(\frac{1}{3}, \frac{2}{3}, \frac{2}{3}\).

Concept Matching

Distance Formula in 2D → \(\sqrt{(x_2-x_1)^2+(y_2-y_1)^2}\)
Slope → \(\frac{y_2-y_1}{x_2-x_1}\)
Circle → Set of points equidistant from a fixed point in a plane
Parabola → Conic with eccentricity 1
Direction Cosines → Cosines of angles made by a line with coordinate axes
Plane → A flat surface represented by \(Ax+By+Cz+D=0\)
Sphere → Set of points equidistant from a fixed point in space
Angle Between Planes → Angle between their normal vectors

Clue Explanation

Analytical geometry questions are solved by identifying the geometrical object and then selecting the correct algebraic representation. For 2D problems, distance, slope, line, circle, and conics are most important. For 3D problems, direction ratios, direction cosines, planes, lines, and spheres are the key areas.

Exam tips

Draw a rough coordinate diagram where needed.
Write coordinates carefully in correct order.
Use distance formula for length-based questions.
Use slope for line direction questions.
Compare circle and sphere equations with standard forms.
Use direction ratios for 3D line equations.
Use normal vectors for plane angle questions.
Revise standard forms of conics regularly.

Practice MCQs

Learning Modules