Mathematics - Straight Lines

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Questions and Answers

What is the equation of the line parallel to the x-axis passing through the point (–2, 3)?

  • x = 3
  • x = –2
  • y = –2
  • y = 3 (correct)

What is the equation of the line parallel to the y-axis at a distance of –2 from the y-axis?

  • y = –2
  • x = 2
  • x = –2 (correct)
  • y = 2

In the point-slope form equation, what does 'm' represent?

  • The slope of the line (correct)
  • The intercept on the y-axis
  • The y-coordinate of the point
  • The x-coordinate of the point

Using the point-slope formula, what is the equation of the line with slope –4 that passes through (–2, 3)?

<p>4x + y + 5 = 0 (B), y – 3 = –4 (x + 2) (C)</p> Signup and view all the answers

Which relationship provides the slope between two points P1 (x1, y1) and P2 (x2, y2)?

<p>y2 – y1 = m(x2 – x1) (B)</p> Signup and view all the answers

If the point-slope equation is given as y – y0 = m(x – x0), what does y0 represent?

<p>The y-coordinate of the fixed point (C)</p> Signup and view all the answers

To find the equation of a line passing through points P1 (–1, 2) and P2 (1, 4), which formula is appropriate?

<p>y – 2 = ((4 – 2)/(1 + 1))(x + 1) (B)</p> Signup and view all the answers

When determining the equation of a line in two-point form, what is the first step?

<p>Identify the slope of the line (D)</p> Signup and view all the answers

How can you demonstrate that points (4, 4), (3, 5), and (–1, –1) form a right-angled triangle?

<p>By calculating the slopes of the lines and showing they are perpendicular. (C)</p> Signup and view all the answers

What is the slope of a line that makes a 30° angle with the positive direction of the y-axis?

<p>$\tan(30°)$ (B)</p> Signup and view all the answers

For the points (x, –1), (2, 1), and (4, 5) to be collinear, what must be the value of x?

<p>3 (D)</p> Signup and view all the answers

How can it be shown that points (–2, –1), (4, 0), (3, 3), and (–3, 2) form a parallelogram?

<p>By showing that opposite sides are parallel and equal in length. (A)</p> Signup and view all the answers

What is the angle between the x-axis and the line joining the points (3,–1) and (4,–2)?

<p>30° (D)</p> Signup and view all the answers

If one line has a slope that is double that of another line, what can be inferred about the angles between them?

<p>The tangent of the angle between them is a fixed ratio. (D)</p> Signup and view all the answers

If the points (h, 0), (a, b), and (0, k) lie on the same line, what relation must hold?

<p>h + k = 1 (C)</p> Signup and view all the answers

In which form can a horizontal line L at a distance 'a' from the x-axis be described?

<p>Both A and C (B)</p> Signup and view all the answers

Flashcards

Slope of a Line

The slope of a line is a measure of its steepness, represented by the ratio of the vertical change (rise) to the horizontal change (run) between any two points on the line.

Perpendicular Lines

Two lines are considered perpendicular if they intersect at a right angle (90 degrees).

Horizontal Line

A line is said to be horizontal if it runs parallel to the x-axis. Its slope is always zero.

Vertical Line

A line is considered vertical if it runs parallel to the y-axis. Its slope is undefined.

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Equation of a Line

The equation of a line is a mathematical expression that defines the relationship between the x and y coordinates of all the points lying on the line. It can take various forms like slope-intercept form, point-slope form, etc.

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Collinear Points

If multiple points lie on the same line, they are said to be collinear. You can check for collinearity by examining if the slopes between pairs of these points are equal.

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Angle Between Two Lines

The angle between two lines is the acute angle formed by the intersection of those two lines. It's determined using the formula: tan(theta) = |(m2 - m1) / (1 + m1 * m2)| where m1 and m2 are the slopes of the lines.

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Point-Slope Form

The point-slope form of a line's equation is expressed as (y - y1) = m(x - x1), where (x1, y1) is a known point on the line and m is the slope of the line.

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Equation of a vertical line

A vertical line at a distance 'b' from the y-axis can be represented by either x = b or x = -b.

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Equation of a horizontal line

A horizontal line at a distance 'a' from the x-axis is represented by the equation y = a.

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Point-slope form equation

The point-slope form of a line with slope 'm' passing through point (x0, y0) is given by: y - y0 = m(x - x0)

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Two-point form

This form is used to find the equation of a line when you know two points it passes through (x1, y1) and (x2, y2).

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Two-point form equation

The two-point form of a line passing through points (x1, y1) and (x2, y2) is: y - y1 = (y2 - y1) / (x2 - x1) * (x - x1)

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Parallel lines

Lines that have the same slope are parallel to each other.

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Study Notes

Mathematics - Straight Lines

  • Right-angled Triangles: Points (4, 4), (3, 5), and (-1, -1) form a right-angled triangle. This can be proven without the Pythagorean theorem.

  • Slope of a Line: A line making a 30° angle with the positive y-axis has a specific slope.

  • Collinear Points: Finding the x-value that makes three points collinear (lie on the same line).

  • Parallelograms: Points (-2, -1), (4, 0), (3, 3), (-3, 2) define the vertices of a parallelogram, which can be shown without using distance formulas.

  • Angle Between Lines: The angle between the x-axis and a line segment connecting (3, -1) and (4, -2) must be calculated.

  • Slopes of Intersecting Lines: If one line's slope is double another and the tangent of the angle between them is 1/2. Calculate the slopes of the lines.

  • Point-Slope Formula: A line passing through (x₁, y₁) and (h, k) with slope m follows a specific equation: k - y₁ = m(h - x₁).

  • Points on a Line: If points lie on a line, 3 points (h,0), (a,b),(0,k) are collinear. It follows a condition.

  • Population Growth: A population graph (Fig 10.10) is given. Calculate the slope of the line AB and predict the population for the year 2010.

Equation of a Line

  • Horizontal and Vertical Lines: Horizontal lines parallel to the x-axis follow equations y = a or y = -a (where 'a' is the distance from the x-axis). Vertical lines parallel to the y-axis follow equations x = b or x = -b (where 'b' is the distance from the y-axis).

  • Point-Slope Form: The equation of a line with slope 'm' passing through point (x₀, y₀) is given by y - y₀ = m(x - x₀).

  • Two-Point Form: The equation of a line passing through points (x₁, y₁) and (x₂, y₂) is y - y₁ = ((y₂-y₁)/(x₂-x₁)) * (x - x₁).

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Mathematics Past Paper PDF - Straight Lines - 2021-22

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