Skip to content
← Business Mathematics and Statistics

Business Mathematics and Statistics · Class 11 Commerce

Ch 3Analytical Geometry (Locus, Straight Lines, Pair of Straight Lines, Circles, Conics) — Class 11 Business Mathematics and Statistics, concept-first.

Analytical geometry lets us study geometric shapes — lines, circles, curves — using algebra and coordinates instead of a compass and ruler alone. Tamil Nadu's Business Mathematics syllabus covers the same coordinate-geometry principles taught nationally, applied here to problems a commerce student will meet again in co…

47

Q&A

8

Concepts

Not available

Exam weightage

Start learning — read this chapter →

Key concepts

Hover a concept to preview it and jump to its most relevant Q&A.

Locus and Its Equation

A locus is the algebraic translation of a geometric rule that a moving point obeys. The working method is always the same: name the moving point , state its distance condition(s) using the distance formula , square to cl…

Most relevant Q&A

Start with this concept →

Chapter contents

The NCERT structure, section by section. Open a section to see its questions, then read the concept-first solution.

1

Introduction to Locus

Analytical geometry lets us study geometric shapes — lines, circles, curves — using algebra and coordinates instead of a compass and ruler alone.

2

Finding the Equation of a Locus

Two conditions come up repeatedly in Business Mathematics problems on locus: a point equidistant from two fixed points, and a point equidistant from a fixed point and a fixed line.

3

Equation of a Straight Line — Standard Forms

A straight line can be written in several equivalent algebraic forms; which one to use depends on what information the problem gives — a point and a slope, two points, or the two axis-intercepts.

4

Angle Between Lines, Parallelism, Perpendicularity and Distance Formulas

Once two lines are written with known slopes and , several useful facts follow directly.

5

Pair of Straight Lines Through the Origin

A single second-degree equation in and can, in a special case, represent not one curve but two straight lines together.

6

Angle Between a Pair of Lines and Special Conditions

For the pair of lines represented by , the angle between them can be found directly from , , without factoring first:

7

General Equation of a Circle

A circle of centre and radius satisfies, by the plain distance definition of a circle (every point on it is at distance from the centre): Expanding this and collecting terms always produces an equatio…

8

Finding the Equation of a Circle from Given Conditions

Depending on what is given, the equation of a circle can be built in different ways, all of which end up in the same general form .

9

Introduction to Conics and the Parabola

A conic section is the curve obtained where a plane cuts a right circular cone; depending on the angle of the cut, the resulting curve is a circle, an ellipse, a parabola, or a hyperbola.

10

Ellipse and Hyperbola — Standard Forms

The ellipse, with eccentricity , has standard equation (major axis along the -axis, centre at the origin): Here is the semi-major axis and the semi-minor axis.

Exercises

+Show 12 questions12 questions
  1. Q3A point moves such that the sum of the squares of its distances from the points $(1,2)$ and $(-1,-2)$ is always $26$. Find the equation of i…Free
  2. Q4A point moves such that its distance from the $x$-axis is always equal to its distance from the point $(0,4)$. Find the equation of the locu…Free
  3. Q7Find the equation of the straight line whose $x$-intercept is $4$ and $y$-intercept is $-3$.Free
  4. Q8Show that the lines $2x+3y-6=0$ and $4x+6y+5=0$ are parallel, and find the distance between them.Preview
  5. Q9Find the angle between the two lines whose slopes are $2$ and $-\dfrac13$, and determine whether the lines are perpendicular.Preview
  6. Q12Find the value of $k$ for which the equation $(k+1)x^2+4xy+(k-1)y^2=0$ represents a pair of perpendicular straight lines.Preview
  7. Q13Find the separate equations of the lines represented by $6x^2-xy-y^2=0$, and find the angle between them.Preview
  8. Q16Find the equation of the circle whose diameter has endpoints $(2,3)$ and $(-4,5)$.Preview
  9. Q17Find the equation of the circle passing through the points $(0,0)$, $(4,0)$, and $(0,3)$.Preview
  10. Q18Determine whether the point $(1,2)$ lies inside, on, or outside the circle $x^2+y^2-4x-6y+9=0$.Preview
  11. Q20For the ellipse $\dfrac{x^2}{25}+\dfrac{y^2}{16}=1$, find $a$, $b$, and the eccentricity $e$.Preview
  12. Q21For the hyperbola $\dfrac{x^2}{9}-\dfrac{y^2}{16}=1$, find $a$, $b$, and the eccentricity $e$.Preview

Sample & Board Papers

Sample papers and previous-year board questions for this subject.

+Show 26 questions26 questions
  1. Q1The double ordinate passing through the focus is : (a) axis (b) focal chord (c) latus rectum (d) directrixPreview
  2. Q2The slope of the line $6x - 2y + 5 = 0$ : (a) $\dfrac{1}{3}$ (b) $3$ (c) $-3$ (d) $0$Preview
  3. Q3Find the length of the tangent from the point $(2, 3)$ to the circle $x^2 + y^2 + 8x + 4y + 8 = 0$.Preview
  4. Q4Find the value of '$a$' for which the straight lines $3x + 4y = 13$; $2x - 7y = -1$ and $ax - y - 14 = 0$ are concurrent.Preview
  5. Q5Show that the equation $2x^2 + 7xy + 3y^2 + 5x + 5y + 2 = 0$ represents a pair of straight lines. Also find the angle between them.Preview
  6. Q6The $x$-intercept of the straight line $3x+2y-1=0$ is : (a) $\dfrac{1}{3}$ (b) $3$ (c) $\dfrac{1}{2}$ (d) $2$Preview
  7. Q7$(1,\ -2)$ is the centre of the circle $x^{2}+y^{2}+ax+by-4=0$, then its radius : (a) $4$ (b) $3$ (c) $1$ (d) $2$Preview
  8. Q8A point in the plane moves so that its distance from the origin is thrice its distance from the $y$-axis. Find its locus.Preview
  9. Q9Find the centre and radius of the circle $6x^{2} + 6y^{2} + 4x - 8y - 16 = 0$.Preview
  10. Q10(a) If the distance of a point from the points $(2,\ 1)$ and $(1,\ 2)$ are in the ratio $2 : 1$, then find the locus of the point. OR (b) Co…Preview
  11. Q11The equation of directrix of the parabola $y^2 = -x$ is : (a) $x-4=0$ (b) $4x+1=0$ (c) $x+4=0$ (d) $4x-1=0$Preview
  12. Q12The angle between the pair of straight lines $x^2 - 7xy + 4y^2 = 0$ is : (a) $\tan^{-1}\left(\dfrac{\sqrt{33}}{5}\right)$ (b) $\tan^{-1}\lef…Preview
  13. Q13Find the focus and vertex of the parabola $y^2 = 20x$.Preview
  14. Q14For what values of a and b does the equation $(a-2)x^2 + by^2 + (b-2)xy + 4x + 4y - 1 = 0$ represents a circle ? Write down the resulting eq…Preview
  15. Q15If $m_1$ and $m_2$ are the slopes of the pair of lines given by $ax^2+2hxy+by^2=0$, then the value of $m_1+m_2$ is : (a) $\dfrac{2h}{a}$ (b)…Preview
  16. Q16Find the equation of the circle with centre at $(4,5)$ and radius 3 units. (a) $x^2+y^2-8x-10y+32=0$ (b) $x^2+y^2-6x+2y-6=0$ (c) $x^2+y^2-8x…Preview
  17. Q17If the centre of the circle $x^2+y^2+2x-6y+1=0$ lies on a straight line $ax+2y+2=0$, then find the value of '$a$'.Preview
  18. Q18Find the values of $a$ and $b$ if the equation $(a-1)x^2+by^2+(b-8)xy+4x+4y-1=0$ represents a circle.Preview
  19. Q19(a) Show that the equation $2x^2+7xy+3y^2+5x+5y+2=0$ represent two straight lines and find their separate equations. OR (b) Solve : $\tan^{-…Preview
  20. Q20(a) Find the equation of the circle passing through the points $(1,0)$, $(-1,0)$ and $(0,1)$. OR (b) Resolve into partial fraction. $\dfrac{…Preview
  21. Q21The angle between the pair of straight lines $x^2-7xy+4y^2=0$ is : (a) $\tan^{-1}\left(\dfrac{\sqrt{33}}{5}\right)$ (b) $\tan^{-1}\left(\dfr…Preview
  22. Q22The equation of the circle with centre on the $x$-axis and passing through the origin is : (a) $x^2+y^2=a^2$ (b) $x^2-2ax+y^2=0$ (c) $x^2-2a…Preview
  23. Q23If $(-2,-7)$ is one extremity of a diameter of the circle $x^2+y^2-2x+6y-15=0$, find the other extremity. (Compulsory)Preview
  24. Q24Find the co-ordinates of the focus, vertex, equation of the directrix of the parabola $x^2=8y$Preview
  25. Q25Find the value of $k$ so that the line $3x+4y-k=0$ is a tangent to the circle $x^2+y^2-64=0$.Preview
  26. Q26(a) Show that the pair of straight lines $4x^2+12xy+9y^2-6x-9y+2=0$ represents two parallel straight lines and also find the separate equati…Preview

More questions

+Show 9 questions9 questions
  1. Example 1Find the equation of the locus of a point which is equidistant from the points $A(2,3)$ and $B(6,-1)$.Free
  2. Example 2Find the equation of the locus of a point which is equidistant from the point $S(3,0)$ and the line $x = -3$.Free
  3. Example 5Find the equation of the straight line passing through the point $(2,-3)$ with slope $4$.Free
  4. Example 6Find the equation of the straight line passing through the points $(1,2)$ and $(3,8)$.Preview
  5. Example 10Show that $2x^2+5xy+2y^2=0$ represents a pair of straight lines, find the separate equations of the two lines, and find the angle between th…Preview
  6. Example 11Determine whether $x^2-4xy+4y^2=0$ represents two distinct lines or two coincident lines.Preview
  7. Example 14Find the centre and radius of the circle $x^2+y^2-6x+8y-11=0$.Preview
  8. Example 15Find the equation of the circle with centre $(2,-3)$ and radius $5$.Preview
  9. Example 19For the parabola $y^2=12x$, find the coordinates of the focus, the equation of the directrix, and the length of the latus rectum.Preview