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Straight Lines

Action Plan Notes Questions Mock-Test

Study Action Plan (Sorted by Difficulty)

Equation of a Line

Accuracy: 25%

Total Attempted4

Total Correct1

Total Incorrect3

Difficulty Level:

High

Distance Between Parallel Lines

Accuracy: 33%

Total Attempted3

Total Correct1

Total Incorrect2

Difficulty Level:

High

Foot of Perpendicular and Altitude Problems

Accuracy: 50%

Learning Objectives

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Learning Objectives

Understand the concept of the slope of a line as the tangent of the angle it makes with the positive x-axis and calculate it using the difference in y-coordinates divided by the difference in x-coordinates.
Derive and apply various forms of the equation of a line, including point-slope form, slope-intercept form, two-point form, and intercept form.
Calculate the perpendicular distance from a point to a line using the formula involving the absolute value of $Ax_1 + By_1 + C$ divided by the square root of $A^2 + B^2$ .
Determine conditions for parallelism and perpendicularity of lines based on their slopes, recognizing that parallel lines have equal slopes and perpendicular lines have slopes whose product is -1.
Compute the angle between two intersecting lines using the formula for the tangent of the angle, given by the absolute difference of the slopes divided by 1 plus the product of the slopes.
Find the distance between two parallel lines using the absolute difference of their constants divided by the square root of $A^2 + B^2$ .
Verify the collinearity of points by ensuring the slope between any two pairs of points is the same.
Transform the general equation of a line $Ax + By + C = 0$ into standard forms like slope-intercept or intercept form, and identify special cases.
Analyze lines cutting the coordinate axes, including equal intercepts, intercept sum/product conditions, and line segments between axes.
Recall coordinate geometry formulas such as the distance formula, section formula, midpoint formula, and area of triangles/quadrilaterals, using zero area for collinearity.
Solve problems involving the foot of the perpendicular from a point to a line, altitude of a triangle, right bisector of a segment, and perpendicular from the origin.
Investigate concurrency and intersection of lines by finding intersection points and checking concurrence of three lines.

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Revision Notes & Summary

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Chapter Notes

Slope of a Line

Definition: The slope of a line is defined as the tangent of the angle it makes with the positive x-axis.
Formula: For a line passing through two points $(x_1, y_1)$ and $(x_2, y_2)$ , the slope $m$ is given by: $m = \frac{y_2 - y_1}{x_2 - x_1}$
Special Cases:
- Horizontal line: $m = 0$
- Vertical line: $m$ is undefined

Equation of a Line

Point-Slope Form: For a line with slope $m$ passing through a point $(x_0, y_0)$ : $y - y_0 = m(x - x_0)$
Slope-Intercept Form: For a line with slope $m$ and y-intercept $c$ : $y = mx + c$
Two-Point Form: For a line passing through points $(x_1, y_1)$ and $(x_2, y_2)$ : $y - y_1 = \frac{y_2 - y_1}{x_2 - x_1}(x - x_1)$
Intercept Form: For a line making intercepts $a$ and $b$ on the x- and y-axes: $\frac{x}{a} + \frac{y}{b} = 1$

Distance from a Point to a Line

Formula: The perpendicular distance $d$ from a point $(x_1, y_1)$ to the line $Ax + By + C = 0$ is given by: $d = \frac{|Ax_1 + By_1 + C|}{\sqrt{A^2 + B^2}}$

Parallel and Perpendicular Lines

Parallel Lines: Two lines are parallel if their slopes are equal, i.e., $m_1 = m_2$ .
Perpendicular Lines: Two lines are perpendicular if the product of their slopes is $-1$ , i.e., $m_1 \cdot m_2 = -1$ .

Angle Between Two Lines

Formula: The tangent of the angle $\theta$ between two lines with slopes $m_1$ and $m_2$ is given by: $\tan \theta = \left| \frac{m_2 - m_1}{1 + m_1 m_2} \right|$

Distance Between Parallel Lines

Formula: For two parallel lines $Ax + By + C_1 = 0$ and $Ax + By + C_2 = 0$ , the distance $d$ between them is: $d = \frac{|C_1 - C_2|}{\sqrt{A^2 + B^2}}$

Collinearity of Points

Condition: Three points $(x_1, y_1)$ , $(x_2, y_2)$ , and $(x_3, y_3)$ are collinear if the slope between any two pairs of points is the same.

General Equation of a Line and Reduction to Standard Forms

General Form: $Ax + By + C = 0$
Reduction: Convert to slope-intercept form $y = mx + c$ or intercept form $\frac{x}{a} + \frac{y}{b} = 1$ by solving for $y$ or finding intercepts.

Intercepts and Lines Cutting Coordinate Axes

Intercepts: The x-intercept is the point where the line crosses the x-axis, and the y-intercept is where it crosses the y-axis.
Equal Intercepts: If a line cuts equal intercepts on the axes, then $a = b$ in the intercept form.

Coordinate Geometry Recall and Area-Based Problems

Distance Formula: The distance between two points $(x_1, y_1)$ and $(x_2, y_2)$ is: $\sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2}$
Area of Triangle: For vertices $(x_1, y_1)$ , $(x_2, y_2)$ , $(x_3, y_3)$ , the area is: $\frac{1}{2} |x_1(y_2-y_3) + x_2(y_3-y_1) + x_3(y_1-y_2)|$

Foot of Perpendicular and Altitude Problems

Foot of Perpendicular: The foot of the perpendicular from a point to a line can be found using the distance formula and the equation of the line.

Concurrency and Intersection of Lines

Intersection: The intersection point of two lines can be found by solving their equations simultaneously.
Concurrency: Three lines are concurrent if they intersect at a single point.

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Practice Test – MCQs, True/False

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Multiple Choice Questions

4

5

8

\frac{8}{\sqrt{5}}

Correct Answer: D

Solution:

The distance between two parallel lines

y = mx + c_1

and

y = mx + c_2

is given by

\frac{|c_2 - c_1|}{\sqrt{1 + m^2}}

. Here,

m = 2

c_1 = 3

c_2 = -5

. Thus, the distance is

\frac{|-5 - 3|}{\sqrt{1 + 4}} = \frac{8}{\sqrt{5}}

Chapter Concept:

Slope of a Line

0°

45°

90°

180°

Correct Answer: C

Solution:

The product of the slopes

m_1 \times m_2 = 2 \times -\frac{1}{2} = -1

, indicating the lines are perpendicular, thus the angle is 90°.

Chapter Concept:

Angle Between Two Lines

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Straight Lines

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True or False

Correct Answer: True

Solution:

The slope of a line is given by

m = \tan \Theta

, where

\Theta

is the angle of inclination with the x-axis. The slope is undefined when

\Theta = 90°

Chapter Concept :

Equation of a Line

Correct Answer: True

Solution:

If the area of a triangle formed by three points is zero, it implies that the points lie on a single straight line, hence they are collinear.

Chapter Concept :

Collinearity of Points

Correct Answer: True

Solution:

The intercept form of a line is given by

\frac{x}{a} + \frac{y}{b} = 1

. If the intercepts are equal, then

a = b

, leading to the equation

\frac{x}{a} + \frac{y}{a} = 1

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The Redemption Arena

Straight Lines

Study Action Plan (Sorted by Difficulty)

Equation of a Line

High

Distance Between Parallel Lines

High

Foot of Perpendicular and Altitude Problems

Learning Objectives

Learning Objectives

Practice, Analyze & Improve 🚀

Revision Notes & Summary

Chapter Notes

Slope of a Line

Equation of a Line

Distance from a Point to a Line

Parallel and Perpendicular Lines

Angle Between Two Lines

Distance Between Parallel Lines

Collinearity of Points

General Equation of a Line and Reduction to Standard Forms

Intercepts and Lines Cutting Coordinate Axes

Coordinate Geometry Recall and Area-Based Problems

Foot of Perpendicular and Altitude Problems

Concurrency and Intersection of Lines

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Practice Test – MCQs, True/False

Multiple Choice Questions

Q1.What is the distance between the parallel lines y=2x+3y = 2x + 3y=2x+3 and y=2x−5y = 2x - 5y=2x−5?

Correct Answer: D

Q2.If the slopes of two lines are m1=2m_1 = 2m1​=2 and m2=−12m_2 = -\frac{1}{2}m2​=−21​, what is the angle between the two lines?

Correct Answer: C

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True or False

Q1.The slope of a line is defined as the tangent of its inclination angle with the positive direction of the x-axis, except when the angle is 90°.

Correct Answer: True

Q2.Three points are collinear if the area of the triangle formed by them is zero.

Correct Answer: True

Q3.A line that makes equal intercepts on the x-axis and y-axis has the equation xa+ya=1\frac{x}{a} + \frac{y}{a} = 1ax​+ay​=1.

Correct Answer: True

Medium

Slope of a Line

Medium

Collinearity of Points

Medium

Concurrency and Intersection of Lines

Low

Distance from a Point to a Line

Low

General Equation of a Line and Reduction to Standard Forms

Low

Angle Between Two Lines

Low

Coordinate Geometry Recall and Area-Based Problems

Low

Intercepts and Lines Cutting Coordinate Axes

Low

Parallel and Perpendicular Lines

Low

Q3.Find the equation of a line that is equidistant from the parallel lines 3x−4y+7=03x - 4y + 7 = 03x−4y+7=0 and 3x−4y+5=03x - 4y + 5 = 03x−4y+5=0.

Correct Answer: A

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Q4.Find the equation of a line that passes through the point (2,−3)(2, -3)(2,−3) and has a slope of 444.

Correct Answer: A

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Q5.A line passes through the point (1,2)(1, 2)(1,2) and is perpendicular to the line 3x+4y−5=03x + 4y - 5 = 03x+4y−5=0. What is the equation of this line?

Correct Answer: A

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Q6.Consider a triangle with vertices at points A(1,2)A(1, 2)A(1,2), B(4,6)B(4, 6)B(4,6), and C(x,y)C(x, y)C(x,y). If the area of the triangle is zero, what is the equation relating xxx and yyy for point CCC?

Correct Answer: A