the distance between the pair of parallel lines x^2+2xy+y^2-6x-6y+8=0 is-------------
Well, the distance between parallel lines is pretty easy to find. You just need to find the perpendicular distance from one line to the other. But let me warn you, this answer might make you parallel park a little differently from now on. So, hold onto your seat!
To find the distance between parallel lines, we need to first rewrite the given equation into a more convenient form. Let's complete the square:
(x^2 + 2xy + y^2) - 6x - 6y + 8 = 0
(x + y)^2 - 6(x + y) + 8 = 0
Now, let's introduce a variable 'd' for the perpendicular distance between the lines.
(x + y)^2 - 6(x + y) + 8 = d^2
Okay, now this is where the clowning around comes in. We can rewrite the equation as:
(x + y)^2 - (6x + 6y) + 8 = d^2
It's like we're having a party and the variables 'x' and 'y' are invited, but they decided to come as a pair - 'x + y.' So, mixing things up a little, we're now going to invite another guest to the party and call it 'D' (or 'd' squared), just because we feel like it.
But, hey, we can't throw a party without some clowning around, right? So, let's go one step further:
(x + y)^2 - 2(3x + 3y) + 8 = d^2
The clown in me just couldn't resist dividing 6 by 2 because, you know, it's nice to have a break once in a while. So we have:
[(x + y)^2] - 2[(3x + 3y)] + 2^2 = d^2
Now, let's rearrange it:
[(x + y)^2] - 2[(x + y)(3)] + 2^2 = d^2
[(x + y - 2)^2] = d^2
Voila! We've eliminated the 'x + y' equation and squared both sides to get a beautiful end result. The equation suggests that the distance between the parallel lines is equal to the square root of 'd^2,' which is just 'd.' In other words...
The distance between the pair of parallel lines is 'd' units!
To find the distance between a pair of parallel lines, we can use the formula:
d = |c1 - c2| / √(a^2 + b^2)
Given the equation of the line as x^2 + 2xy + y^2 - 6x - 6y + 8 = 0, we can rewrite it as:
(x + y - 4)^2 - 16 = 0
Expanding and simplifying:
x^2 + y^2 + 2xy - 8x - 8y + 16 - 16 = 0
x^2 + y^2 + 2xy - 8x - 8y = 0
Now we can compare this equation to the general form of the equation of a line, Ax + By + C = 0, where A = 1, B = 1, and C = 0.
We have:
a = 1, b = 1, c1 = -8, and c2 = -8
Plugging these values into the formula, we get:
d = |c1 - c2| / √(a^2 + b^2)
= |-8 - (-8)| / √(1^2 + 1^2)
= 0 / √2
= 0
Therefore, the distance between the pair of parallel lines x^2 + 2xy + y^2 - 6x - 6y + 8 = 0 is 0.
To find the distance between two parallel lines, we need to find the perpendicular distance between them.
The given equation is x^2 + 2xy + y^2 - 6x - 6y + 8 = 0. Let's rewrite it in a clearer form:
x^2 + 2xy + y^2 - 6x - 6y + 8 = (x + y - 4)(x + y - 2) = 0
Now, we have two lines:
Line 1: x + y - 4 = 0
Line 2: x + y - 2 = 0
Notice that the coefficients of 'x' and 'y' in both equations are equal. This indicates that the lines are parallel.
To find the distance between these parallel lines, we can use the formula for the perpendicular distance between a point and a line. In this case, we will consider one of the lines as the reference line and find the distance between the other line and that reference line.
Take Line 1 (x + y - 4 = 0) as the reference line and consider the other line (Line 2: x + y - 2 = 0).
We can find the distance using the following steps:
1. Find the perpendicular distance between the point (a, b) and Line 1.
2. Substitute the coordinates of the point (a, b) in Line 2: x + y - 2 = 0.
3. Solve the equation to find the value of 'a' or 'b'.
4. Use the distance formula with (a, b) and Line 1 to find the perpendicular distance.
Without additional information, we cannot proceed any further in finding the coordinates of the point (a, b) or calculating the distance between the parallel lines.
The two lines are
y = 2-x
y = 4-x
So the distance between them is 2/sin45°