Three Dimensional Geometry

To determine the values of for which the equation represents a pair of distinct lines, we proceed:

1. Recognizing the Equation Type:
The given equation is a homogeneous equation of degree 2, suggesting it may represent a pair of lines passing through the origin.

2. Condition for Distinct Lines:
For a homogeneous quadratic equation to represent a pair of distinct lines, the discriminant condition must be satisfied: . Identify the coefficients by comparing with the standard form:

3. Computing the Discriminant:
Calculate the discriminant:


For distinct lines, we need:

4. Solving the Inequality:
Solve the quadratic inequality . First, find the roots of the equation :


Roots are and . Since the parabola opens upward, test the sign in the intervals , , and :

- For :
- For :
- For :
Thus, when or .

5. Expressing the Solution:
The solution to the inequality is:


This can be written as:

Final Answer:
The equation represents a pair of distinct lines for all .

To solve the problem, we need to find the sum of the squares of the side lengths of triangle ABC given the midpoints of its sides.

1. Define the Vertices and Midpoints:
Let the vertices of triangle ABC be A = (x₁, y₁, z₁), B = (x₂, y₂, z₂), C = (x₃, y₃, z₃).
Let D, E, F be the midpoints of BC, CA, AB respectively, given as D = (3, 0, 0), E = (0, 4, 0), F = (0, 0, 5).

2. Set Up Midpoint Equations:
For D = ((x₂+x₃)/2, (y₂+y₃)/2, (z₂+z₃)/2) = (3, 0, 0):
x₂ + x₃ = 6, y₂ + y₃ = 0, z₂ + z₃ = 0.
For E = ((x₃+x₁)/2, (y₃+y₁)/2, (z₃+z₁)/2) = (0, 4, 0):
x₃ + x₁ = 0, y₃ + y₁ = 8, z₃ + z₁ = 0.
For F = ((x₁+x₂)/2, (y₁+y₂)/2, (z₁+z₂)/2) = (0, 0, 5):
x₁ + x₂ = 0, y₁ + y₂ = 0, z₁ + z₂ = 10.

3. Solve for x-Coordinates:
From x₁ + x₂ = 0, x₁ = -x₂.
From x₃ + x₁ = 0, x₃ = -x₁ = x₂.
From x₂ + x₃ = 6, x₂ + x₂ = 6, so 2x₂ = 6, x₂ = 3.
Thus, x₁ = -3, x₃ = 3.

4. Solve for y-Coordinates:
From y₁ + y₂ = 0, y₁ = -y₂.
From y₃ + y₁ = 8, y₃ = 8 - y₁ = 8 + y₂.
From y₂ + y₃ = 0, y₂ + 8 + y₂ = 0, so 2y₂ = -8, y₂ = -4.
Thus, y₁ = 4, y₃ = 4.

5. Solve for z-Coordinates:
From z₁ + z₂ = 10, z₁ = 10 - z₂.
From z₃ + z₁ = 0, z₃ = -z₁ = z₂ - 10.
From z₂ + z₃ = 0, z₂ + z₂ - 10 = 0, so 2z₂ = 10, z₂ = 5.
Thus, z₁ = 5, z₃ = -5.

6. Determine the Vertices:
The vertices are A = (-3, 4, 5), B = (3, -4, 5), C = (3, 4, -5).

7. Calculate the Side Lengths Squared:
AB² = (3 - (-3))² + (-4 - 4)² + (5 - 5)² = 6² + (-8)² = 36 + 64 = 100.
BC² = (3 - 3)² + (4 - (-4))² + (-5 - 5)² = 0 + 8² + (-10)² = 64 + 100 = 164.
CA² = (-3 - 3)² + (4 - 4)² + (5 - (-5))² = (-6)² + 0 + 10² = 36 + 100 = 136.

8. Sum the Squares:
AB² + BC² + CA² = 100 + 164 + 136 = 400.

Final Answer:
The sum of the squares of the side lengths is 400.

Step 1: Understanding the Problem:
We are given that the number of diagonals of a polygon is 35. The number of diagonals of a polygon with sides is given by the formula:

Step 2: Solving for :
We are told that the number of diagonals is 35, so we can substitute this into the formula:


Multiplying both sides by 2:

Step 3: Simplifying the Equation:
Expanding the left-hand side:


Rearranging the equation:

Step 4: Solving the Quadratic Equation:
We will solve the quadratic equation using the quadratic formula:



First, simplify inside the square root:

Step 5: Finding :
Taking the two possible solutions:


or

Step 6: Conclusion:
The number of sides of the polygon is , as the number of sides cannot be negative.

Step 7: Finding the Number of Triangles:
To find the number of triangles that can be formed by selecting three vertices of the polygon, we use the combination formula:

Number of triangles =
Substituting :

Step 8: Using the Given Condition:
We are given that and are two distinct vertices of the polygon, and we need to find the number of triangles that include as one of their sides. To form such a triangle, we need to choose one more vertex from the remaining 8 vertices.
Thus, the number of such triangles is 8.

Final Answer:
The number of triangles formed by joining three vertices of the polygon with as one of its sides is 8.

We are given a triangle with sides in the ratio 4:5:6. Let's set , , and for some .

1. Calculate the Semi-perimeter:

2. Compute the Area using Heron's Formula:

3. Find the Inradius:
Using , we get:

4. Calculate the Circumradius:
Using :

5. Compute the Ratio of Circumradius to Inradius:

Final Answer:
The ratio of the circumradius to the inradius is

To find the vertex of the triangle formed by the lines , , and

1. Finding the First Vertex:
Solve for the intersection of and :


Add the equations:



Substitute into (2):


Thus, the intersection is .

2. Verifying Other Vertices:
Solve for the intersection of and :



Add the equations:



Substitute into (1):


So, the intersection is .

3. Third Vertex:
Solve for the intersection of and :



Subtract (1) from (2):



Substitute into (1):


So, the intersection is .


However, using makes all three lines concurrent at , which does not form a triangle.
Thus, the original line is likely correct, as it yields as a vertex.

Final Answer:
The vertex is .

The correct option is: (A) π/6