The equation of a line that is perpendicular to the given line 4x - 12y = 2 and that passes through the point (10, -1) is 3x + y = 29
"The slope-intercept form of equation of line is, y = mx + c where m is the slope and c is the y-intercept."
"The equation of the line passing through point (p. q) and having slope 'm' is, (y - q) = m(x - p)."
For given question,
We have been given an equation of the line.
4x - 12y = 2
We write above equation in slope-intercept form.
⇒ 4x - 12y = 2
⇒ -12y = -4x + 2
[tex]\Rightarrow y=\frac{1}{3}x-\frac{1}{6}[/tex]
From the above equation, the slope of the line 4x - 12y = 2 is, [tex]m_1=\frac{1}{3}[/tex]
We know, if two lines are perpendicular, then the product of their slopes is −1.
Let [tex]m_2[/tex] be the slope of the required line.
The required line is perpendicular to the given line.
[tex]\Rightarrow m_1\times m_2=-1\\\\\Rightarrow \frac{1}{3}\times m_2=-1\\\\ \Rightarrow m_2=-3[/tex]
Also, the required line passes though the point (10, -1)
Using slope-point form of equation of line,
[tex]\Rightarrow (y - q) = m_2(x - p)\\\\\Rightarrow (y-(-1))=(-3)\times (x-10)\\\\\Rightarrow (y+1)=-3x+30\\\\\Rightarrow y=-3x+29\\\\\Rightarrow 3x + y=29[/tex]
Therefore, the equation of a line that is perpendicular to the given line 4x - 12y = 2 and that passes through the point (10, -1) is 3x + y = 29
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