find the equation of the planes 5x-3y+4=0 and x+y-2z+10=0 and which are 1 unit distance from the origin.?
To find the equation of planes that are 1 unit distance from the origin, we can use the formula for the distance between a point and a plane. The general equation for a plane is "Ax + By + Cz + D = 0" where A, B, C represent the coefficients of x, y, z respectively, and D represents the constant term.
Let's start with the first plane, 5x - 3y + 4 = 0.
Step 1: Rewrite the equation in the standard form:
5x - 3y + 4 = 0
5x - 3y = -4
Step 2: The coefficients A, B, C of the plane equation are:
A = 5
B = -3
C = 0 (since there is no z term)
Step 3: Use the formula for the distance between a point and a plane, which is:
distance = |Ax + By + Cz + D| / sqrt(A^2 + B^2 + C^2)
Since the plane is 1 unit distance from the origin, the distance in the formula is 1. Therefore:
1 = |5(0) - 3(0) + 0D + (-4)| / sqrt(5^2 + (-3)^2 + 0^2)
Simplifying, we get:
1 = 4 / sqrt(34)
Step 4: Solve for D:
4 / sqrt(34) = |D|
Since D represents the constant term in the plane equation and it must be multiplied by -4, it means D can be either 4 / sqrt(34) or -4 / sqrt(34).
If D = 4 / sqrt(34), then the equation of the first plane is:
5x - 3y + 4 = 0
If D = -4 / sqrt(34), then the equation of the first plane is:
5x - 3y - 4 = 0
Now, let's find the equation of the second plane, x + y - 2z + 10 = 0.
Step 1: Rewrite the equation in the standard form:
x + y - 2z + 10 = 0
Step 2: The coefficients A, B, C of the plane equation are:
A = 1
B = 1
C = -2
Step 3: Use the formula for the distance between a point and a plane, which is:
distance = |Ax + By + Cz + D| / sqrt(A^2 + B^2 + C^2)
Since the plane is 1 unit distance from the origin, the distance in the formula is 1. Therefore:
1 = |1(0) + 1(0) - 2(0) + D| / sqrt(1^2 + 1^2 + (-2)^2)
Simplifying, we get:
1 = |D| / sqrt(6)
Step 4: Solve for D:
|D| = sqrt(6)
Since D represents the constant term in the plane equation and it must be multiplied by -1, it means D can be either sqrt(6) or -sqrt(6).
If D = sqrt(6), then the equation of the second plane is:
x + y - 2z + 10 = 0
If D = -sqrt(6), then the equation of the second plane is:
x + y - 2z - 10 = 0
So, we have two possible equations for each plane given the condition of being 1 unit distance from the origin:
First plane:
1) 5x - 3y + 4 = 0
2) 5x - 3y - 4 = 0
Second plane:
3) x + y - 2z + 10 = 0
4) x + y - 2z - 10 = 0