find the equation of the line which satisfy the given condition: Intersecting the y-axis at a distance of 2 units above the origin and making an angle of 30o with positive direction of the x-axis.

Class 11 | Exercise 10.2 | Maths | NCERT | Straight Lines

Given: \[\mathbf{\theta }\text{ }=\text{ }\mathbf{30}{}^\circ \]

We realize that incline, \[\mathbf{m}\text{ }=\text{ }\mathbf{tan}\text{ }\mathbf{\theta }\]

\[\mathbf{m}\text{ }=\text{ }\mathbf{tan30}{}^\circ \text{ }=\text{ }\left( \mathbf{1}/\surd \mathbf{3} \right)\]

We realize that the point (x, y) on the line with incline m and y-catch c lies on the line if and provided that \[\mathbf{y}\text{ }=\text{ }\mathbf{mx}\text{ }+\text{ }\mathbf{c}.\]

In case distance is 2 units over the beginning, \[\mathbf{c}\text{ }=\text{ }+\mathbf{2}\]

Thus, \[\mathbf{y}\text{ }=\text{ }\left( \mathbf{1}/\surd \mathbf{3} \right)\mathbf{x}\text{ }+\text{ }\mathbf{2}\]

\[\mathbf{y}\text{ }=\text{ }\left( \mathbf{x}\text{ }+\text{ }\mathbf{2}\surd \mathbf{3} \right)/\surd \mathbf{3}\]

\[\surd \mathbf{3}\text{ }\mathbf{y}\text{ }=\text{ }\mathbf{x}\text{ }+\text{ }\mathbf{2}\surd \mathbf{3}\]

\[\mathbf{x}\text{ }\text{ }\surd \mathbf{3}\text{ }\mathbf{y}\text{ }+\text{ }\mathbf{2}\surd \mathbf{3}\text{ }=\text{ }\mathbf{0}\]

∴ The condition of the line is \[\mathbf{x}\text{ }\text{ }\surd \mathbf{3}\text{ }\mathbf{y}\text{ }+\text{ }\mathbf{2}\surd \mathbf{3}\text{ }=\text{ }\mathbf{0}.\]

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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.

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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