![\boxed{\sf a = 9 }](/tpl/images/1395/0988/86aa0.png)
Step-by-step explanation:
Two lines are given to us which are perpendicular to each other and we need to find out the value of a . The given equations are ,
![\sf\longrightarrow 3x - 2y +7=0](/tpl/images/1395/0988/4b8be.png)
![\sf\longrightarrow 6x +ay -18 = 0](/tpl/images/1395/0988/8af4b.png)
Step 1 : Convert the equations in slope intercept form of the line .
![\sf\longrightarrow y = \dfrac{3x}{2} +\dfrac{ 7 }{2}](/tpl/images/1395/0988/47735.png)
and ,
![\sf\longrightarrow y = -\dfrac{6x }{a}+\dfrac{18}{a}](/tpl/images/1395/0988/252fa.png)
Step 2: Find the slope of the lines :-
Now we know that the product of slope of two perpendicular lines is -1. Therefore , from Slope Intercept Form of the line we can say that the slope of first line is ,
![\sf\longrightarrow Slope_1 = \dfrac{3}{2}](/tpl/images/1395/0988/bf845.png)
And the slope of the second line is ,
![\sf\longrightarrow Slope_2 =\dfrac{-6}{a}](/tpl/images/1395/0988/d9a39.png)
Step 3: Multiply the slopes :-
![\sf\longrightarrow \dfrac{3}{2}\times \dfrac{-6}{a}= -1](/tpl/images/1395/0988/effdb.png)
Multiply ,
![\sf\longrightarrow \dfrac{-9}{a}= -1](/tpl/images/1395/0988/117c7.png)
Multiply both sides by a ,
![\sf\longrightarrow (-1)a = -9](/tpl/images/1395/0988/08331.png)
Divide both sides by -1 ,
![\sf\longrightarrow \boxed{\blue{\sf a = 9 }}](/tpl/images/1395/0988/d1237.png)
Hence the value of a is 9 .