Question

find the equation of line passing through A(1,2) B(1,-2)

Answer 1

$\boxed{\underline{\underline{\bf{\star\:Answer}}}}\\\\\\\textsf{y+4x-6=0}\\\\\\\boxed{\underline{\underline{\bf{\star\:Step-by-step\: explanation}}}}\\\\\\\underline{\textsf{Given}}\\\\\textsf{Point\:A\:=\{1,2\}}\\\textsf{Point\:A\:= \{1,2\}}\\\\\textsf{So,}\\\textsf{ x_1=1 }\\\textsf{ y_1=2 }\\\textsf{ x_2=1 }\\\textsf{ y_2=-2 }\\\\\\\boxed{\underline{\underline{\bf{\star\:To\:Calculate}}}}\\\\\\\textsf{Equation\:of\:line\:passing\:through\:A\:and\:B}$

$\boxed{\star\:\underline{\underline{\bf{Steps}}}}\\\\\\\textsf{Slope\:=\: \frac{y_2-y_1}{x_2-x_1} }\\\\\textsf{Substituting\:the\:value\: x_1,x_2,y_1 \:and\: y_2 \:in\:the\:above\:formula}\\\\\textsf{Slope\:=\: \frac{-2-2}{1-1} }\\\Longrightarrow{\textsf{Slope\:=\:-\:4}}$

We know that,

$\texttt{Equation\:of\:line\:passing\:through\:two\:points\:}\\\Longrightarrow{\texttt{y\:-\: y_1 \:=\:m\:(\:x\:-\: x_1 \:)}}\\\\\textsf{Substituting\:values\:of\: x_1,y_1 \:and\:m\:in\:the\:above\:equation}$

y-2=-4(x-1)

y-2=-4x=4

y+4x-6=0

y+4x-6=0 is the required answer.

Some Other Formulas

• Equation of a line parallel to x-axis at a distance b.

          y=b (where b is constant)

• Equation of a line parallel to y-axis at a distance a.

          x=a (where a is constant)

• Equation of line in slope intercept form

          y=mx+c (where m is slope and c is intercept made by line on y-axis)

• Equation of line in point-slope form or one-point form

         $y-y_1=m(x-x_1)$

• Equation of a non-vertical line passing through two given points

        $A(x_1,y_1)\:and\:B(x_2,y_2)\:is$

         $y-y_1=\frac{y_2-y_1}{x_2-x_1}(x-x_1)$

         This is called two-point form.

        $We \:can \:also\:write\:m\:instead\:of\:\frac{y_2-y_1}{x_2-x_1}\:$

• Condition of two lines to be parallel is that the slope of the two lines must be equal.

       $m_1=m_2$

• Condition of two lines to be perpendicular is that the product of slopes of the two lines must be equal to -1.

        $m_1 \times m_2=-1$