find the equation of the line which satisfy the given condition: Find the equation of the line passing through (–3, 5) and perpendicular to the line through the points (2, 5) and (–3, 6).

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

Given:

Focuses are \[\left( \mathbf{2},\text{ }\mathbf{5} \right)\] and \[\left( -\text{ }\mathbf{3},\text{ }\mathbf{6} \right).\]

We realize that slant, \[\mathbf{m}\text{ }=\text{ }\left( \mathbf{y2}\text{ }\text{ }\mathbf{y1} \right)/\left( \mathbf{x2}\text{ }\text{ }\mathbf{x1} \right)\]

\[=\text{ }\left( \mathbf{6}\text{ }\text{ }\mathbf{5} \right)/\left( -\text{ }\mathbf{3}\text{ }\text{ }\mathbf{2} \right)\]

\[=\text{ }\mathbf{1}/\text{ }-\text{ }\mathbf{5}\text{ }=\text{ }-\text{ }\mathbf{1}/\mathbf{5}\]

We realize that two non-vertical lines are opposite to one another if and provided that their slants are negative reciprocals of one another.

Then, at that point, \[\mathbf{m}\text{ }=\text{ }\left( -\text{ }\mathbf{1}/\mathbf{m} \right)\]

\[=\text{ }-\text{ }\mathbf{1}/\left( -\text{ }\mathbf{1}/\mathbf{5} \right)\]

\[=\text{ }\mathbf{5}\]

We realize that the point (x, y) lies on the line with slant m through the proper point (x0, y0), if and provided that, its directions fulfill the condition \[\mathbf{y}\text{ }\text{ }\mathbf{y0}\text{ }=\text{ }\mathbf{m}\text{ }\left( \mathbf{x}\text{ }\text{ }\mathbf{x0} \right)\]

Then, at that point, \[\mathbf{y}\text{ }\text{ }\mathbf{5}\text{ }=\text{ }\mathbf{5}\left( \mathbf{x}\text{ }\text{ }\left( -\text{ }\mathbf{3} \right) \right)\]

\[\mathbf{y}\text{ }\text{ }\mathbf{5}\text{ }=\text{ }\mathbf{5x}\text{ }+\text{ }\mathbf{15}\]

\[\mathbf{5x}\text{ }+\text{ }\mathbf{15}\text{ }\text{ }\mathbf{y}\text{ }+\text{ }\mathbf{5}\text{ }=\text{ }\mathbf{0}\]

\[\mathbf{5x}\text{ }\text{ }\mathbf{y}\text{ }+\text{ }\mathbf{20}\text{ }=\text{ }\mathbf{0}\]

∴ The condition of the line is \[\mathbf{5x}\text{ }\text{ }\mathbf{y}\text{ }+\text{ }\mathbf{20}\text{ }=\text{ }\mathbf{0}\]

i

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.

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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