The slope of a line is double of the slope of another line. If tangent of the angle between them is 1/3, find the slopes of the lines.

Class 11 | Exercise 10.1 | Maths | NCERT | Straight Lines

Allow us to consider ‘m1’ and ‘m’ be the incline of the two given lines to such an extent that

$\mathbf{m1}\text{ }=\text{ }\mathbf{2m}$

We realize that in case θ is the point somewhere within l1 and l2 with incline m1 and m2, then, at that point,

$\mathbf{1}+\mathbf{2m2}\text{ }=\text{ }-\text{ }\mathbf{3m}$

$\mathbf{2m2}\text{ }+\mathbf{1}\text{ }+\mathbf{3m}\text{ }=\text{ }\mathbf{0}$

$\mathbf{2m}\text{ }\left( \mathbf{m}+\mathbf{1} \right)\text{ }+\text{ }\mathbf{1}\left( \mathbf{m}+\mathbf{1} \right)\text{ }=\text{ }\mathbf{0}$

$\left( \mathbf{2m}+\mathbf{1} \right)\text{ }\left( \mathbf{m}+\mathbf{1} \right)=\text{ }\mathbf{0}$

$\mathbf{m}\text{ }=\text{ }-\text{ }\mathbf{1}\text{ }\mathbf{or}\text{ }-\text{ }\mathbf{1}/\mathbf{2}$

In the event that

$\mathbf{m}\text{ }=\text{ }-\text{ }\mathbf{1}$

, the incline of the lines are

$-\text{ }\mathbf{1}\text{ }\mathbf{and}\text{ }-\text{ }\mathbf{2}$

In the event that

$\mathbf{m}\text{ }=\text{ }-\text{ }\mathbf{1}/\mathbf{2}$

, the incline of the lines are

$-\text{ }\mathbf{1}/\mathbf{2}\text{ }\mathbf{and}\text{ }-\text{ }\mathbf{1}$

Case 2:

$\mathbf{2m2}\text{ }\text{ }\mathbf{3m}\text{ }+\text{ }\mathbf{1}\text{ }=\text{ }\mathbf{0}$

$\mathbf{2m2}\text{ }\text{ }\mathbf{2m}\text{ }\text{ }\mathbf{m}\text{ }+\text{ }\mathbf{1}\text{ }=\text{ }\mathbf{0}$

$\mathbf{2m}\text{ }\left( \mathbf{m}\text{ }\text{ }\mathbf{1} \right)\text{ }\text{ }\mathbf{1}\left( \mathbf{m}\text{ }\text{ }\mathbf{1} \right)\text{ }=\text{ }\mathbf{0}$

$\mathbf{m}\text{ }=\text{ }\mathbf{1}\text{ }\mathbf{or}\text{ }\mathbf{1}/\mathbf{2}$

In the event that

$\mathbf{m}\text{ }=\text{ }\mathbf{1}$

, the incline of the lines are 1 and 2

In the event that

$\mathbf{m}\text{ }=\text{ }\mathbf{1}/\mathbf{2}$

, the incline of the lines are 1/2 and 1

∴ The incline of the lines are

$\left[ -\mathbf{1}\text{ }\mathbf{and}\text{ }-\text{ }\mathbf{2} \right]\text{ }\mathbf{or}\text{ }\left[ -\mathbf{1}/\mathbf{2}\text{ }\mathbf{and}\text{ }-\text{ }\mathbf{1} \right]\text{ }\mathbf{or}\text{ }\left[ \mathbf{1}\text{ }\mathbf{and}\text{ }\mathbf{2} \right]\text{ }\mathbf{or}\text{ }\left[ \mathbf{1}/\mathbf{2}\text{ }\mathbf{and}\text{ }\mathbf{1} \right]$

Find the vector equation of the line passing through the point with position vector and parallel to the vector Deduce the Cartesian equations of the line.

A line passes through the point (2, 1, -3) and is parallel to the vector Find the equations of the line in vector and Cartesian forms.

A line passes through the point (3, 4, 5) and is parallel to the vector Find the equations of the line in the vector as well as Cartesian forms.

Evaluate:

If $f(x) = |x| - 3,find\mathop {\lim }\limits_{x \to 3} f(x)$

Let $\begin{array}{l} f(x) = \left\{ \begin{array}{l} \frac{x}{{|x|}},x \ne 0\\ 0,x = 0 \end{array} \right.\\ \end{array}$ . Show that $\begin{array}{l} \mathop {\lim }\limits_{x \to 0} f(x) \end{array}$ does not exist.

Let $\begin{array}{l} f(x) = \left\{ \begin{array}{l} \frac{|x-3|}{{(x-3)'}},x \ne 3\\ 0,x = 3 \end{array} \right.\\ \end{array}$ . Show that $\begin{array}{l} \mathop {\lim }\limits_{x \to 3} f(x) \end{array}$ does not exist.

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More Material

Find the angle between each of the following pairs of lines:

Find the coordinates of the foot of the perpendicular drawn from the point (1, 2, 3) to the line

Show that the lines and do not intersect each other.

Find the vector equation of the line passing through the point with position vector and parallel to the vector Deduce the Cartesian equations of the line.

A line passes through the point (2, 1, -3) and is parallel to the vector Find the equations of the line in vector and Cartesian forms.

A line passes through the point (3, 4, 5) and is parallel to the vector Find the equations of the line in the vector as well as Cartesian forms.

Evaluate:

Evaluate:

Evaluate:

If $f(x) = |x| - 3,find\mathop {\lim }\limits_{x \to 3} f(x)$

Let $\begin{array}{l} f(x) = \left\{ \begin{array}{l} \frac{x}{{|x|}},x \ne 0\\ 0,x = 0 \end{array} \right.\\ \end{array}$ . Show that $\begin{array}{l} \mathop {\lim }\limits_{x \to 0} f(x) \end{array}$ does not exist.

Let $\begin{array}{l} f(x) = \left\{ \begin{array}{l} \frac{|x-3|}{{(x-3)'}},x \ne 3\\ 0,x = 3 \end{array} \right.\\ \end{array}$ . Show that $\begin{array}{l} \mathop {\lim }\limits_{x \to 3} f(x) \end{array}$ does not exist.

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