What is the acute angle between the pair of straight lines \(\sqrt 2 {\rm{x}} + \sqrt 3 {\rm{y}} = 1\) and \(\sqrt 3 {\rm{x}} + \sqrt 2 {\rm{y}} = 2?\)
Understanding the Angle Between Two Straight Lines
This problem asks us to find the acute angle between two given straight lines. The equations of the lines are:
- Line 1: \(\sqrt 2 {\rm{x}} + \sqrt 3 {\rm{y}} = 1\)
- Line 2: \(\sqrt 3 {\rm{x}} + \sqrt 2 {\rm{y}} = 2\)
To find the angle between two lines, we first need to determine their slopes. The general form of a linear equation is \(Ax + By + C = 0\). The slope (\(m\)) of such a line is given by the formula \(m = -\frac{A}{B}\).
Calculating the Slopes of the Straight Lines
Let's find the slopes for each line:
- For Line 1: \(\sqrt 2 {\rm{x}} + \sqrt 3 {\rm{y}} - 1 = 0\)
Here, \(A_1 = \sqrt 2\) and \(B_1 = \sqrt 3\).
The slope \(m_1 = -\frac{A_1}{B_1} = -\frac{\sqrt 2}{\sqrt 3} = -\sqrt{\frac{2}{3}}\). - For Line 2: \(\sqrt 3 {\rm{x}} + \sqrt 2 {\rm{y}} - 2 = 0\)
Here, \(A_2 = \sqrt 3\) and \(B_2 = \sqrt 2\).
The slope \(m_2 = -\frac{A_2}{B_2} = -\frac{\sqrt 3}{\sqrt 2} = -\sqrt{\frac{3}{2}}\).
Applying the Angle Formula Between Two Lines
The formula for the angle \(\theta\) between two lines with slopes \(m_1\) and \(m_2\) is given by:
\(\tan \theta = \left| \frac{m_1 - m_2}{1 + m_1 m_2} \right|\)
This formula gives the tangent of one of the angles between the lines. The absolute value ensures that we get a non-negative value for \(\tan \theta\), which corresponds to the acute angle.
Calculating the terms for the formula:
First, let's calculate the difference in slopes, \(m_1 - m_2\):
\(m_1 - m_2 = -\sqrt{\frac{2}{3}} - \left(-\sqrt{\frac{3}{2}}\right) = -\frac{\sqrt{2}}{\sqrt{3}} + \frac{\sqrt{3}}{\sqrt{2}}\)
To subtract these fractions, we find a common denominator, which is \(\sqrt{3} \cdot \sqrt{2} = \sqrt{6}\):
\(m_1 - m_2 = \frac{(-\sqrt{2})(\sqrt{2}) + (\sqrt{3})(\sqrt{3})}{\sqrt{6}} = \frac{-2 + 3}{\sqrt{6}} = \frac{1}{\sqrt{6}}\)
Next, let's calculate the product of the slopes, \(m_1 m_2\):
\(m_1 m_2 = \left(-\sqrt{\frac{2}{3}}\right) \left(-\sqrt{\frac{3}{2}}\right) = \left(\sqrt{\frac{2}{3}}\right) \left(\sqrt{\frac{3}{2}}\right) = \sqrt{\frac{2}{3} \cdot \frac{3}{2}} = \sqrt{1} = 1\)
Now, let's calculate the denominator of the angle formula, \(1 + m_1 m_2\):
\(1 + m_1 m_2 = 1 + 1 = 2\)
Substituting the values into the angle formula:
Now we substitute the calculated values into the formula for \(\tan \theta\):
\(\tan \theta = \left| \frac{\frac{1}{\sqrt{6}}}{2} \right| = \left| \frac{1}{2\sqrt{6}} \right|\)
Since \(\frac{1}{2\sqrt{6}}\) is positive, the absolute value does not change the result:
\(\tan \theta = \frac{1}{2\sqrt{6}}\)
Finding the Acute Angle
To find the angle \(\theta\), we take the inverse tangent (arctangent) of the value we found:
\(\theta = \tan^{-1}\left(\frac{1}{2\sqrt{6}}\right)\)
Since we used the absolute value in the formula, this angle \(\theta\) is the acute angle between the two lines.
Let's compare this result with the given options:
- Option 1: \({\tan ^{ - 1}}\left( {\frac{1}{{2\sqrt 6 }}} \right)\)
- Option 2: \({\tan ^{ - 1}}\left( {\frac{1}{{\sqrt 2 }}} \right)\)
- Option 3: tan -1 (3)
- Option 4: \({\tan ^{ - 1}}\left( {\frac{1}{{\sqrt 3 }}} \right)\)
Our calculated acute angle matches Option 1.
Summary of Steps
- Identify the equations of the two straight lines.
- Calculate the slope of each line using the formula \(m = -A/B\).
- Use the formula \(\tan \theta = \left| \frac{m_1 - m_2}{1 + m_1 m_2} \right|\) to find the tangent of the angle between the lines.
- Calculate \(\theta\) by taking the inverse tangent: \(\theta = \tan^{-1} \left(\left| \frac{m_1 - m_2}{1 + m_1 m_2} \right|\right)\).
- The result is the acute angle between the lines.
| Line Equation | Form \(Ax + By + C = 0\) | Slope (\(m = -A/B\)) |
|---|---|---|
| \(\sqrt 2 {\rm{x}} + \sqrt 3 {\rm{y}} = 1\) | \(\sqrt 2 {\rm{x}} + \sqrt 3 {\rm{y}} - 1 = 0\) | \(m_1 = -\frac{\sqrt 2}{\sqrt 3}\) |
| \(\sqrt 3 {\rm{x}} + \sqrt 2 {\rm{y}} = 2\) | \(\sqrt 3 {\rm{x}} + \sqrt 2 {\rm{y}} - 2 = 0\) | \(m_2 = -\frac{\sqrt 3}{\sqrt 2}\) |
Revision Table: Angle Between Lines
| Concept | Description | Formula |
|---|---|---|
| Slope of a line \(Ax+By+C=0\) | Measure of the steepness of the line. | \(m = -A/B\) |
| Angle \(\theta\) between two lines with slopes \(m_1, m_2\) | The angle formed at the intersection of the two lines. Acute angle is obtained using absolute value. | \(\tan \theta = \left| \frac{m_1 - m_2}{1 + m_1 m_2} \right|\) |
| Inverse Tangent (\(\tan^{-1}\)) | Function that gives the angle whose tangent is a given number. | If \(\tan \theta = x\), then \(\theta = \tan^{-1}(x)\). |
Additional Information: Angle Between Lines
When two lines intersect, they form two pairs of angles: a pair of acute angles and a pair of obtuse angles. The formula \(\tan \theta = \left| \frac{m_1 - m_2}{1 + m_1 m_2} \right|\) always gives a non-negative value for \(\tan \theta\), which corresponds to the acute angle \(\theta\) because the tangent of an angle between \(0^\circ\) and \(90^\circ\) is positive.
If the lines are parallel, their slopes are equal (\(m_1 = m_2\)), and the numerator \(m_1 - m_2\) is 0, leading to \(\tan \theta = 0\), so \(\theta = 0^\circ\). Parallel lines do not intersect, or they are the same line; the angle between them is considered \(0^\circ\).
If the lines are perpendicular, the product of their slopes is -1 (\(m_1 m_2 = -1\)). In this case, the denominator \(1 + m_1 m_2 = 1 + (-1) = 0\), making \(\tan \theta\) undefined. An undefined tangent corresponds to an angle of \(90^\circ\), which is the definition of perpendicular lines.
The given line equations are linear equations in two variables \(x\) and \(y\). These equations represent straight lines in the Cartesian coordinate system.
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