The general equation of the plane through a line is given by (3+k)x + (2-3k)y + z(1+2k) -2-13k, for some integer k
How do we find the equation of the line containing above and parallel to the line given by,
(x+5)/3 = (y+4)/1 = (z-7)/-2
My initial thought was to take proportions of the two lines' direction ratios, but instead the answer given by the professor has multiplied the direction ratios and then their sums has been equated to zero, as in,
3(3+k) + 1(2-3k)y - 2(1+2k) = 0
Can I know the reason for that?
To find the equation of a line parallel to the given line and passing through the plane, we need to find the direction ratios of the line. One way to do this is by taking ratios of the coefficients of x, y, and z in the equations of the two lines. However, in this case, it seems that your professor followed a slightly different approach.
Your professor multiplied the direction ratios (coefficients of x, y, and z) of the given line by the sum of their squares and set it equal to zero. Let's break down the steps to understand the reasoning behind it:
1. Start with the equation of the plane through the line:
(3+k)x + (2-3k)y + (1+2k)z - 2 - 13k = 0
2. Write down the direction ratios of the given line:
(x+5)/3 = (y+4)/1 = (z-7)/-2
The direction ratios of the given line are 3, 1, and -2.
3. Multiply the direction ratios by the sum of their squares:
(3+k) * 3 + (2-3k) * 1 + (1+2k) * (-2) = 0
Let's simplify this equation step by step:
9 + 3k + 2 - 3k - 2 - 4k = 0
11 - 4k = 0
4. Solve for k:
11 = 4k
k = 11/4
5. Finally, substitute the value of k into the equation of the plane from step 1 to get the equation of the line:
(3 + 11/4)x + (2 - 3(11/4))y + (1 + 2(11/4))z - 2 - 13(11/4) = 0
(12/4 + 11/4)x + (8/4 - 33/4)y + (4/4 + 22/4)z - 8/4 - 143/4 = 0
(23/4)x + (-25/4)y + (26/4)z - 151/4 = 0
Multiplying through by 4 to get rid of fractions:
23x - 25y + 26z - 151 = 0
Therefore, the equation of the line parallel to the given line and passing through the plane is 23x - 25y + 26z - 151 = 0.
So, the reason your professor multiplied the direction ratios by the sum of their squares and equated it to zero was to determine the value of k (which is 11/4 in this case) and subsequently derive the equation of the line.