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How To Solve: Slope of a Line
[#permalink]
15 May 2023, 09:24
15 May 2023, 09:24
1
Show: :: DOWNLOAD PDF: How To Solve: Slope of a Line
Attachment:
How To Solve: Slope of a Line
Attached pdf of this Article as SPOILER at the top! Happy learning!
Hi All,
I have recently uploaded a video on YouTube to discuss Slope of a Line in Detail:
Following is covered in the video
- ¤ Proof of Slope of a Line formula
¤ How to find slope of a Generic Line
¤ Sign of Slope of a Line
¤ Slope of Parallel and Perpendicular Lines
¤ Solved Problem : Find slope of a line given two points
¤ Solved Problem : Find slope of a line given equation of a line
¤ Solved Problem : Find value of a variable given the slope of a line
¤ Solved Problem : Find equation of a line parallel to a line and passing through a point
¤ Solved Problem : Find equation of a line perpendicular to a line and passing through a point
Proof of Slope of a Line formula
Let's say we have a line(l) passing through two point A(\(x_1\),\(y_1\)) and B(\(x_2\),\(y_2\)) and having a slope of m.
Lets draw perpendicular lines as shown in the below figure to complete △ABC
Let ∠BAC = θ
Attachment:
Slope of a Line - 1.jpg [ 16.96 KiB | Viewed 1728 times ]
Slope of a line, m = tan(Angle made by the line with positive X-Axis) = tan(θ) = Opposite Side / Adjacent Side = \(\frac{BC}{AC}\) = \(\frac{y_2 - y_1}{x_2 - x_1}\)
How to find slope of a Generic Line
Let's say we have a generic equation of the line given by ax + by = c
=> by = -ax + c
=> y = \(\frac{-a}{b}\)x + \(\frac{c}{b}\)
Comparing above equation with generic equation of a line y = mx + b we get
m = \(\frac{-a}{b}\)
=> Slope of the line ax + by = c, is given by m = \(\frac{-a}{b}\) = -Coefficient of x / Coefficient of y
Sign of Slope of a Line
¤ Positive Slope: Line tilted towards right
¤ Negative Slope: Line tilted towards left
¤ Zero Slope: Line parallel to x-axis
¤ Infinite Slope: Line parallel to y-axis
Attachment:
Sign of slope.jpg [ 22.08 KiB | Viewed 1751 times ]
Slope of Parallel and Perpendicular Lines
If two lines are Parallel, then their slopes will be equal.
¤ If we have two parallel lines \(L_1\) and \(L_2\), with below equations
¤ Line L1 : y = \(m_1\)x + \(c_1\)
¤ Line L2 : y = \(m_2\)x + \(c_2\)
¤ then \(m_1\) = \(m_2\)
Attachment:
Parallel Lines.jpg [ 4.93 KiB | Viewed 1715 times ]
If two lines are Perpendicular, then product of their slopes will be equal to -1
¤ If we have two perpendicular lines \(L_1\) and \(L_2\), with below equations
¤ Line L1 : y = \(m_1\)x + \(c_1\)
¤ Line L2 : y = \(m_2\)x + \(c_2\)
¤ then \(m_1\) * \(m_2\) = -1
Attachment:
Perpendicular Lines.jpg [ 3.41 KiB | Viewed 1695 times ]
Solved Problem : Find slope of a line given two points
Q1. Find the slope of the line passing through two points (1,3) and (3,5).
Show: ::
Sol 1: Slope of a line is given by m = \(\frac{y_2 - y_1}{x_2 - x_1}\) = \(\frac{5 - 3}{3 - 1}\)
= \(\frac{2}{2}\) = 1
= \(\frac{2}{2}\) = 1
Solved Problem : Find slope of a line given equation of a line
Q2. Find the slope of the line 2x + 3y = 5.
Show: ::
Sol 2: Slope of the line ax + by = c, is given by m = -Coefficient of x / Coefficient of y
= \(\frac{-2}{3}\)
= \(\frac{-2}{3}\)
Solved Problem : Find value of a variable given the slope of a line
Q3. A line with slope 2 passes through the points (x,5) and (2,x). Find the value of x.
Show: ::
Sol 3: Slope of a line is given by m = \(\frac{y_2 - y_1}{x_2 - x_1}\)
= \(\frac{x - 5}{2 - x}\) = 2 (given)
=> x - 5 = 4 - 2x
=> 3x = 9
=> x = \(\frac{9}{3}\) = 3
= \(\frac{x - 5}{2 - x}\) = 2 (given)
=> x - 5 = 4 - 2x
=> 3x = 9
=> x = \(\frac{9}{3}\) = 3
Solved Problem : Find equation of a line parallel to a line and passing through a point
Q4. Find the equation of the line which is parallel to the line 2x + 4y = 5 and passes through the point (2,3).
Show: ::
Sol 4: Equation of line 2x + 4y = 5 = \(\frac{-2}{4}\) = \(\frac{-1}{2}\)
=> Parallel line will have the same slope = \(\frac{-1}{2}\)
=> Equation of the line is given by y - \(y_1\) = m * (x - \(x_1\))
=> y - 3 = \(\frac{-1}{2}\) * (x - 2)
=> 2y - 6 = -x + 2
=> x + 2y = 8 is the equation of the line parallel to 2x + 4y = 5 and passes through the point (2,3)
=> Parallel line will have the same slope = \(\frac{-1}{2}\)
=> Equation of the line is given by y - \(y_1\) = m * (x - \(x_1\))
=> y - 3 = \(\frac{-1}{2}\) * (x - 2)
=> 2y - 6 = -x + 2
=> x + 2y = 8 is the equation of the line parallel to 2x + 4y = 5 and passes through the point (2,3)
Solved Problem : Find equation of a line perpendicular to a line and passing through a point
Q5. Find the equation of the line which is perpendicular to the line 2x + 4y = 5 and passes through the point (2,3).
Show: ::
Sol 5: Equation of line 2x + 4y = 5 = \(\frac{-2}{4}\) = \(\frac{-1}{2}\)
=> Slope of perpendicular line, m will be given by m * \(\frac{-1}{2}\) = -1
=> m = 2
=> Equation of the line is given by y - \(y_1\) = m * (x - \(x_1\))
=> y - 3 = \([fraction]2[/fraction]\) * (x - 2)
=> y - 3 = 2x - 4
=> y = 2x - 1 is the equation of the line perpendicular to 2x + 4y = 5 and passes through the point (2,3)
=> Slope of perpendicular line, m will be given by m * \(\frac{-1}{2}\) = -1
=> m = 2
=> Equation of the line is given by y - \(y_1\) = m * (x - \(x_1\))
=> y - 3 = \([fraction]2[/fraction]\) * (x - 2)
=> y - 3 = 2x - 4
=> y = 2x - 1 is the equation of the line perpendicular to 2x + 4y = 5 and passes through the point (2,3)
Watch the following video to learn the Basics of Co-ordinate Geometry
Watch the following video to learn about Equation of a Line
Hope it helps!
Good Luck!
Re: How To Solve: Slope of a Line
[#permalink]
23 Nov 2024, 14:15
23 Nov 2024, 14:15
Expert Reply
Thanks for this marvelous topic sir
gmatclubot
Re: How To Solve: Slope of a Line [#permalink]
23 Nov 2024, 14:15
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