How to Use This Page
This page provides complete solutions for the Analytical Geometry chapter from the Samacheer Kalvi 11th Business Maths textbook. Follow this approach for maximum benefit:
1. Attempt every question on your own first. Work through each problem with pen and paper before looking at the solution. 2. Compare your answer with the solution given here. If your answer matches, move on. If it differs, rework the problem step by step. 3. Focus on the method, not just the final answer. The step-by-step working is what builds your understanding and earns marks in exams. 4. Pay special attention to the Common Mistakes section — these are the most frequent errors students make in board exams. 5. Use the Exam Checklist before your test to ensure you have covered all key topics.
Tip: For numerical problems, always write the formula first, then substitute values. This earns partial credit even if your final arithmetic is wrong.
Key Concepts (Quick Recap)
Distance Formula
| Formula | Description | ||
|---|---|---|---|
d = √[(x₂−x₁)² + (y₂−y₁)²] | Distance between two points (x₁,y₁) and (x₂,y₂) | ||
| `d = | x₂−x₁ | ` | Distance between two points on the x-axis (y₁ = y₂) |
| `d = | y₂−y₁ | ` | Distance between two points on the y-axis (x₁ = x₂) |
Section Formula
| Formula | Description |
|---|---|
P = ((mx₂+nx₁)/(m+n), (my₂+ny₁)/(m+n)) | Internal division in ratio m:n |
P = ((mx₂−nx₁)/(m−n), (my₂−ny₁)/(m−n)) | External division in ratio m:n |
M = ((x₁+x₂)/2, (y₁+y₂)/2) | Midpoint formula (ratio 1:1) |
Slope Formula
| Formula | Description |
|---|---|
m = (y₂−y₁)/(x₂−x₁) | Slope of line through (x₁,y₁) and (x₂,y₂) |
m = tan θ | Slope in terms of inclination angle θ |
m = −A/B | Slope from general equation Ax + By + C = 0 |
Equations of a Straight Line
| Form | Equation | What You Need |
|---|---|---|
| Slope-Intercept | y = mx + c | Slope m, y-intercept c |
| Point-Slope | y − y₁ = m(x − x₁) | Slope m, one point (x₁, y₁) |
| Two-Point | (y−y₁)/(y₂−y₁) = (x−x₁)/(x₂−x₁) | Two points on the line |
| Intercept | x/a + y/b = 1 | x-intercept a, y-intercept b |
| General | Ax + By + C = 0 | Standard form for all lines |
Parallel and Perpendicular Lines
| Condition | Meaning |
|---|---|
m₁ = m₂ | Lines are parallel |
m₁ · m₂ = −1 | Lines are perpendicular |
| Slopes equal, intercepts different | Distinct parallel lines |
| Slopes equal, intercepts same | Coincident lines (same line) |
Angle Between Two Lines
Distance of a Point from a Line
Area of a Triangle
I. Choose the Best Answer (MCQs)
Question 1
The slope of the line joining the points (2, 3) and (6, 7) is:
(a) 1 (b) 2 (c) 4 (d) 1/2
The slope is 1, so the line rises one unit vertically for every one unit horizontally.
Question 2
The distance between the points (1, 2) and (4, 6) is:
(a) 3 (b) 5 (c) 7 (d) 25
Question 3
The midpoint of the line segment joining (4, 6) and (2, 8) is:
(a) (3, 7) (b) (6, 14) (c) (1, 1) (d) (2, 2)
Question 4
If two lines have slopes m₁ = 2 and m₂ = −1/2, then the lines are:
(a) Parallel (b) Perpendicular (c) Coincident (d) Intersecting but not perpendicular
Question 5
The equation of a line with slope 3 and y-intercept 4 is:
(a) y = 3x − 4 (b) y = 4x + 3 (c) y = 3x + 4 (d) y = −3x + 4
Question 6
The point dividing the line segment joining (1, 2) and (5, 6) in the ratio 2:1 internally is:
(a) (11/3, 14/3) (b) (3, 4) (c) (7/3, 10/3) (d) (9/3, 12/3)
Question 7
The distance of the point (3, 4) from the line 3x + 4y − 5 = 0 is:
(a) 1 (b) 4 (c) 2 (d) 7/5
II. Fill in the Blanks
1. The slope of a horizontal line is 0.
A horizontal line has the form y = constant. Since there is no change in y, slope = (change in y)/(change in x) = 0.
2. The slope of a vertical line is undefined (or infinity).
A vertical line has the form x = constant. Since x₁ = x₂, the denominator in the slope formula becomes zero, making the slope undefined.
| 3. The distance of the point (0, 0) from the line Ax + By + C = 0 is ** | C | / √(A² + B²)**. |
|---|
| Substituting (0, 0) into the point-line distance formula: d = | A(0) + B(0) + C | / √(A² + B²) = | C | / √(A² + B²). |
|---|
4. The area of the triangle formed by the points (0, 0), (a, 0), and (0, b) is ab/2.
| Using the area formula: Area = ½ | 0(0 − b) + a(b − 0) + 0(0 − 0) | = ½ | ab | = ab/2. |
|---|
5. If the slope of a line is 1, its inclination with the positive x-axis is 45°.
Since m = tan θ, tan θ = 1 implies θ = 45°.
6. The equation of the x-axis is y = 0.
Every point on the x-axis has y-coordinate equal to 0.
7. The equation of the y-axis is x = 0.
Every point on the y-axis has x-coordinate equal to 0.
8. Two lines are parallel if their slopes are equal.
Parallel lines never meet; they have the same steepness (m₁ = m₂).
9. Two lines are perpendicular if the product of their slopes is −1.
This comes from the condition tan θ is undefined when 1 + m₁m₂ = 0, i.e., m₁m₂ = −1.
10. The centroid of a triangle with vertices (x₁, y₁), (x₂, y₂), (x₃, y₃) is ((x₁+x₂+x₃)/3, (y₁+y₂+y₃)/3).
The centroid divides each median in the ratio 2:1, yielding the average of the vertex coordinates.
11. The distance between the origin and the point (3, 4) is 5.
d = √[(3−0)² + (4−0)²] = √(9 + 16) = √25 = 5.
12. If the area of a triangle formed by three points is zero, then the three points are collinear.
Zero area means all three points lie on the same straight line, so no triangle is formed.
III. True or False
State whether each statement is True or False. If False, provide the correction.
1. Statement: The distance between two points is always positive.
True. The distance formula involves a square root, which is always non-negative. When the two points are distinct, the distance is strictly positive.
2. Statement: The slope of a line parallel to the y-axis is 0.
False. The slope of a line parallel to the y-axis (a vertical line) is undefined, not 0. A line parallel to the x-axis (a horizontal line) has slope 0. Correction: The slope of a line parallel to the y-axis is undefined (or infinite).
3. Statement: The point (2, 3) lies on the line 2x + 3y = 13.
True. Substituting x = 2 and y = 3: 2(2) + 3(3) = 4 + 9 = 13. The point satisfies the equation.
4. Statement: The section formula for external division gives a point between the two given points.
False. External division gives a point outside the line segment, beyond one of the endpoints. Internal division gives a point between the two given points. Correction: The section formula for external division gives a point outside the line segment.
5. Statement: If two lines have slopes m₁ and m₂ such that m₁ = −1/m₂, then the lines are perpendicular.
True. Rearranging: m₁ · m₂ = −1, which is the condition for perpendicular lines.
IV. Match the Following
| Column A | Column B |
|---|---|
| 1. Slope of line through (1, 1) and (3, 5) | (e) 2 |
| 2. Distance between (0, 0) and (6, 8) | (d) 10 |
| 3. Midpoint of (2, 4) and (6, 10) | (a) (4, 7) |
| 4. Equation of line with slope 2, y-intercept 1 | (f) y = 2x + 1 |
| 5. Condition for parallel lines | (b) m₁ = m₂ |
| 6. Condition for perpendicular lines | (c) m₁ · m₂ = −1 |
Answers with working:
1 → (e): m = (5−1)/(3−1) = 4/2 = 2
2 → (d): d = √(36 + 64) = √100 = 10
3 → (a): M = ((2+6)/2, (4+10)/2) = (4, 7)
4 → (f): y = mx + c → y = 2x + 1
5 → (b): Parallel lines have equal slopes: m₁ = m₂
6 → (c): Perpendicular lines satisfy m₁ · m₂ = −1
V. Answer Briefly
Question 1
Define the slope of a line and write the formula for the slope of a line passing through two given points.
The slope of a line measures its steepness or inclination relative to the positive x-axis. It is the ratio of the change in y-coordinate (rise) to the change in x-coordinate (run) between any two points on the line.
For a line passing through points (x₁, y₁) and (x₂, y₂):
The slope can be positive (line rises left to right), negative (line falls left to right), zero (horizontal line), or undefined (vertical line). The slope also equals the tangent of the angle the line makes with the positive x-axis: m = tan θ, where θ is the inclination.
Question 2
State the section formula for internal division and explain its significance.
The section formula for internal division states that if a point P divides the line segment joining A(x₁, y₁) and B(x₂, y₂) internally in the ratio m:n, then:
This formula is significant because it allows us to find the coordinates of any point on a line segment without drawing it. When m = n (ratio 1:1), it reduces to the midpoint formula:
The section formula is widely used in geometry, physics, and engineering to locate points of division, centres of mass, and weighted averages.
Question 3
Write the various forms of the equation of a straight line.
The equation of a straight line can be expressed in several forms, each suited to different given conditions:
| Form | Equation | When to Use |
|---|---|---|
| Slope-Intercept | y = mx + c | When slope m and y-intercept c are known |
| Point-Slope | y − y₁ = m(x − x₁) | When slope m and one point (x₁, y₁) are known |
| Two-Point | (y−y₁)/(y₂−y₁) = (x−x₁)/(x₂−x₁) | When two points on the line are known |
| Intercept | x/a + y/b = 1 | When x-intercept a and y-intercept b are known |
| General | Ax + By + C = 0 | Standard form used for all line equations |
Each form can be converted to any other form through algebraic manipulation. The general form is the most versatile and is required for computing distances from a point to a line.
Question 4
What is the condition for three points to be collinear? Explain with the area formula.
Three points (x₁, y₁), (x₂, y₂), and (x₃, y₃) are collinear if and only if the area of the triangle they form equals zero:
When the area is zero, the three points lie on the same straight line — no triangle is formed. This is because the height of the "triangle" becomes zero when all points are aligned. Alternatively, collinearity can be checked by verifying that the slope from point 1 to point 2 equals the slope from point 2 to point 3:
Both methods must give the same result.
VI. Answer in Detail
Since the y-intercept is 4, we know the line passes through (0, 4). We also know the line passes through (2, 3).
Step 1: Find the slope.
Step 2: Write the equation using slope-intercept form.
Multiplying both sides by 2 to eliminate the fraction:
Step 3: Find the x-intercept.
The x-intercept is found by setting y = 0:
Step 1: Apply the section formula for internal division.
Step 2: Find the distance of P from the origin (0, 0).
Calculating further:
Step 1: Find the slopes of both lines.
Step 2: Check if the lines are perpendicular.
Since m₁ · m₂ = −1, the lines are perpendicular.
Step 3: State the angle.
The angle between the lines is 90°.
Step 4: Find the point of intersection by solving simultaneously.
Multiply (i) by 2 and (ii) by 3:
Adding (iii) and (iv):
Substituting x = 11/13 into (i):
Common Mistakes to Avoid
1. Confusing slope-intercept form with point-slope form. Remember: y = mx + c uses the y-intercept c, while y − y₁ = m(x − x₁) uses a point (x₁, y₁). Do not substitute the intercept into the point-slope form or vice versa.
| 2. Forgetting absolute value in distance formulas. Both the distance between two points and the distance from a point to a line must be positive. Always use the absolute value: `d = | Ax₀ + By₀ + C | / √(A² + B²)`. |
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3. Mixing up internal and external division. Internal division uses + in both numerators: (mx₂ + nx₁)/(m + n). External division uses −: (mx₂ − nx₁)/(m − n). Always check whether the dividing point lies between or outside the given points.
| 4. Sign errors in the area of triangle formula. The formula is `Area = ½ | x₁(y₂ − y₃) + x₂(y₃ − y₁) + x₃(y₁ − y₂) | `. Do not forget the absolute value, and pay close attention to the order of subtractions — each y term is subtracted from the next cyclically. |
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5. Assuming undefined slope means zero slope. A vertical line has undefined slope (division by zero), not zero. A horizontal line has slope zero. This distinction is critical when checking parallel/perpendicular conditions.
| 6. Not converting to general form before using point-to-line distance formula. The formula `d = | Ax₀ + By₀ + C | /√(A² + B²) requires the line in the form Ax + By + C = 0. If your equation is 2x + 3y = 7, rewrite it as 2x + 3y − 7 = 0` before substituting A, B, C. |
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| 7. Forgetting the denominator in the angle formula. The formula is `tan θ = | (m₁ − m₂)/(1 + m₁m₂) | . If 1 + m₁m₂ = 0`, the lines are perpendicular — do not attempt to divide by zero. In that case, θ = 90° directly. |
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8. Incorrectly identifying collinearity due to rounding. Three points are collinear only if the area of the triangle is exactly zero. Rounding errors can make a near-zero value appear to be zero. Always simplify fully before concluding collinearity.
Exam Checklist
Use this checklist before your exam. Mark each item once you are confident.
- [ ] I can compute the slope between any two points using
m = (y₂ − y₁)/(x₂ − x₁) - [ ] I know all five forms of the equation of a straight line and when to use each one
- [ ] I can apply the distance formula between two points and from a point to a line
- [ ] I can use the section formula for both internal and external division
- [ ] I can derive the midpoint formula as a special case of section formula (ratio 1:1)
- [ ] I can check if two lines are parallel (
m₁ = m₂) or perpendicular (m₁ · m₂ = −1) - [ ] I can find the angle between two lines using
tan θ = |(m₁ − m₂)/(1 + m₁m₂)| - [ ] I can calculate the area of a triangle given three vertices and check collinearity
- [ ] I can find the point of intersection of two lines by solving simultaneous equations
- [ ] I remember to use absolute values in distance calculations and never confuse slope zero with undefined slope