Answer :
To solve this problem, we need to find both the value of the constant [tex]k[/tex] and the slope of the line [tex]l_1[/tex]. Let's address each part of the question step-by-step:
(i) Finding the value of [tex]k[/tex]:
The equation of the line [tex]l_1[/tex] is given as:
[tex]2x - 3y + k = 0[/tex]
We know that the point [tex]A(2, 3)[/tex] lies on this line. This means that when [tex]x = 2[/tex] and [tex]y = 3[/tex], the equation of the line should be satisfied.
Substitute [tex]x = 2[/tex] and [tex]y = 3[/tex] into the equation:
[tex]2(2) - 3(3) + k = 0[/tex]
Simplify the equation:
[tex]4 - 9 + k = 0[/tex]
Combine the numbers:
[tex]-5 + k = 0[/tex]
Solve for [tex]k[/tex] by adding 5 to both sides:
[tex]k = 5[/tex]
So, the value of [tex]k[/tex] is 5.
(ii) Finding the slope of [tex]l_1[/tex]:
The general form of the equation for a line is [tex]Ax + By + C = 0[/tex]. The slope [tex]m[/tex] of the line given by this equation can be calculated using the formula [tex]m = -\frac{A}{B}[/tex].
In this case, [tex]A = 2[/tex] and [tex]B = -3[/tex]. Therefore, the slope [tex]m[/tex] of the line [tex]l_1[/tex] is:
[tex]m = -\frac{2}{-3} = \frac{2}{3}[/tex]
Thus, the slope of [tex]l_1[/tex] is [tex]\frac{2}{3}[/tex].
In summary:
- The value of [tex]k[/tex] is 5.
- The slope of [tex]l_1[/tex] is [tex]\frac{2}{3}[/tex].
