1. Find the equation to the cone which passes through the three coordinte axes and the lines x/1=y/-2=z/3andx/5=y/1=z/1
Answers
Answer:
The general equation of a cone with vertex at origin is given by
ax2+by2+cz2+2fyz+2gzx+2hxy=0
It is given that the three coordinate axes are generating line of this cone. Therefore the direction number of the of these lines will satisfy the equation of the cone.
The direction number of x-axis is (1,0,0), from here we will get that a=0.
Similarly, we will get that b=0 and c=0.
Therefore now the equation of the required cone is reduced to
2fyz+2gzx+2hxy=0
Since the cone passes through two more lines and there direction number is given, so we will get the following two equations:
−6f+3g−2h=0
and
−2f−3g+6h=0
Step-by-step explanation:
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The equation of the cone cannot be formed.
Given:
The equation to the cone which passes through the three coordinate axes and the lines x/1 = y/-2 = z/3 and x/5 = y/1 = z/1
Solution:
The general equation of a cone is ax² +by² +cz² +2fyz + 2gzx + 2hxy = 0
Let ax² +by² +cz² +2fyz + 2gzx + 2hxy = 0 --- (1) be the equation of cone
From the data,
The cone passes throgh 3 coordinate axes that are are x, y, and z-axes
As we know the coordinates points on the x, y, and z axes are (x, 0, 0)
(0, y, 0) and (0, 0, z)
Equation (1) will be written as
At (x, 0, 0)
=> ax² +b(0)² +c(0)² +2f(0)z + 2g(0)x + 2h(0)x = 0
=> ax² = 0
=> a = 0
At (0, y, 0)
=> a(0)² +by² +c(0)² +2fy(0) + 2g(0)(0) + 2h(0)y = 0
=> by² = 0
=> b = 0
At (0, 0, z)
=> a(0)² +b(0)² +cz² +2f(0)z + 2gz(0) + 2h(0)y = 0
=> cz² = 0
=> c = 0
Hence, a = 0, b = 0, and c = 0
The equation of the cone is 2fyz + 2gzx + 2hxy = 0
=> fyz + gzx + hxy = 0 ---- (2)
Given that the cone is passes the lines x/1 = y/-2 = z/3 and x/5 = y/1 = z/1
Their direction coordinates are (1, -2, 3) and (5, 1, 1)
From (1, -2, 3)
=> f(-2)(3) + g(3)(1) + h(1)(-2) = 0
=> -6f + 3g - 2h = 0 ----- (3)
From (5, 1, 1)
=> f(1(1)+ gz(5) + h(1)(1) = 0
=> f + 5g + h = 0 ---- (4)
Here the resultant expressions are
-6f + 3g - 2h = 0 and f + 5g + h = 0 which cannot be solved
Hence,
The equation of the cone cannot be formed.
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