The equation of a line passing through the point \(\left( { - 4,3} \right)\), and perpendicular to line joining the points \(\left( {4,1} \right)\) and \(\left( { - 4, - 3} \right)\) is \({\bf{y - 3 =-2\left( {x + 4} \right)}\).
Further explanation:
The equation of the line passing through point \(\left( {{x_1},{y_1}} \right)\) and \(\left( {{x_2},{y_2}} \right)\) is as follows:
\(y - {y_1} = \dfrac{{{y_2} - {y_1}}}{{{x_2} - {x_1}}}\left( {x - {x_1}} \right)\) ……(1)
The given line passing through the points \(\left( {4,1} \right)\) and \(\left( { - 4, - 3} \right)\).
Consider the point \(\left( {4,1} \right)\) as \(\left( {{x_1},{y_1}} \right)\) and \(\left( { - 4, - 3} \right)\) as \(\left( {{x_2},{y_2}} \right)\).
Substitute 4 for \({x_1}\), 1 for \({y_1}\), -4 for \({x_2}\) and -3 for \({y_2}\) in equation (1).
\(\begin{aligned}y - 1 &= \frac{{ - 3 - 1}}{{ - 4 - 4}}\left( {x - 4} \right) \hfill \\y - 1& = \frac{{ - 4}}{{ - 8}}\left( {x - 4} \right) \hfill \\y - 1 &= \frac{1}{2}\left( {x - 4} \right) \hfill \\ \end{aligned}\)
The point slope form of the equation passing through point \(\left( {{x_1},{y_1}} \right)\) is as follows:
\(y - {y_1} = m\left( {x - {x_1}} \right)\) ……(2)
Here, \(m\) is the slope of the line.
Compare the equation (2) with \(y - 1 = \dfrac{1}{2}\left( {x - 4} \right)\) to obtain the slope \({m_1}\) of line.
Thus, the slope \({m_1}\) of the first line is \(\dfrac{1}{2}\).
The slope \({m_2}\) of the other line perpendicular to first line is obtained as follows:
\({m_1} \times {m_2} =-1\) ……(3)
Substitute \(\dfrac{1}{2}\) for \({m_1}\) in equation (3).
\(\begin{aligned}\frac{1}{2} \cdot {m_2} &=-1 \hfill \\{m_2} &=-2 \hfill \\ \end{aligned}\)
Since, the second line passing through point \(\left( { - 4,3} \right)\), therefore, the equation of the second line is obtained as follows:
Substitute -4 for \({x_1}\), 3 for \({y_1}\) and -2 for \(m_2\) in equation (2).
\(\begin{aligned}y - 3 &=-2\left( {x - \left( { - 4} \right)} \right) \hfill \\y - 3 &=-2\left( {x + 4} \right) \hfill \\ \end{aligned}\)
Thus, the equation of the line passing through point \(\left( { - 4,3} \right)\) and perpendicular to line \(y - 1 = \dfrac{1}{2}\left( {x - 4} \right)\) is \({\bf{y - 3 =-2\left( {x + 4} \right)}\).
The graph is attached below in figure 1.
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3. what are the domain and range of the function f(x) = 3x + 5?
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Answer Details :
Grade: Junior high School.
Subject: Mathematics.
Chapter: Coordinate geometry.
Keywords:
lines, straight lines, (4,1),function, point slope, equation of line passing, perpendicular line, graph, domain, intervals, intercepts, function value, intercepts of lines, slope, slope intercept form, continuous, range, point, line segment.
