Straight Lines

We are given the equation of a line in point-slope form: where and are fixed, and different lines are drawn by varying .

Step 1: General form of the line
Rearranging the equation: This shows that the value of shifts the line vertically for each different value of , while the slope and remain fixed.

Step 2: Analysis of the lines
Since is fixed, all lines will have the same slope and therefore will be parallel to each other.

Step 3: Intersection with the line
If we substitute into the equation of the line: Thus, each line will intersect the vertical line at a point , where is the specific value for each line.

Answer:

Given:
Equation:

Step 1: Rewrite the expression
Group the terms:
So the given expression becomes:

Step 2: Apply difference of squares:

So either:

Step 3: Equation of line:
Let the general equation of a line be:
We now test the points (2, 3) and (-2, -3) in this line.

Case 1: Let (2, 3) satisfy the line:
This matches one of the factors found above.

Case 2: Let (-2, -3) satisfy the line:
Which is equivalent to: Again, this matches the other factor found.

Conclusion:
The line passes through both points (2, 3) and (-2, -3).

Answer: Option (A): (2, 3) or (–2, –3)

Three lines are concurrent if they intersect at a single point. For the lines , , and to be concurrent, the determinant of their coefficients must be zero:

Expanding the determinant:

We are given that the line passes through one of the points , , , or .

Let's test each point:

1. If the line passes through :

. This matches the equation we derived from the concurrency condition.

2. If the line passes through :

. This does not match.

3. If the line passes through :

. This does not match.

4. If the line passes through :

. This does not match.

The only point that satisfies the equation is .

Total possible combinations: Each ring has 15 letters: Number of unsuccessful attempts: Check (A): Does 482 divide N? Check if: Check (C): Is N the product of 3 distinct prime numbers? Prime factorization of 3374: Factor 1687: All are distinct primes, so yes.
Correct options: (A) ✅ 482 divides N (C) ✅ N is the product of three distinct primes

The coordinates of the rectangle's vertices are A(1, 1), B(1, 2), C(x, y), and D(-1, -1). Side AB lies along the line , and side AD is parallel to the y-axis.

Let's compute the slope of AB using the coordinates A(1, 1) and B(1, 2):

Slope of AB = , which is undefined. This means AB is a vertical line, so the x-coordinates of points A, B, and C must be equal.

Therefore, point C must also lie on the vertical line to complete the rectangle.

Now, we are told that side AB lies along the line , which implies that both points A and B must lie on that line.

Check for point A(1, 1):
, → So A does not lie on the line

Check for point B(1, 2):
, → So B lies on the line

That means only point B lies on the line , which suggests the question might intend side BC to lie along , not AB.

Let's suppose side BC lies on the line , and we want to find the coordinate C(x, y) such that:

• C lies on the line
• CD is parallel to AB, so it must also be vertical
• Hence, x-coordinate of C must be equal to that of B, which is 1

Substituting into , we get:
→ But B is already (1, 2), so C can't be the same as B.

Therefore, the correct values for C must satisfy both the line equation and the rectangle’s symmetry, assuming D is known to be (-1, -1).

Using the midpoint formula (since diagonals of a rectangle bisect each other):

Let C = (x, y).
Midpoint of AC =
Midpoint of BD =

Equating midpoints:

So, one possible point is C = (-1, 0), but this point does not lie on .
Try other values: for example, test C = (3, 8) → → Not equal.
Try C = (-3, -1) → → Not equal either.

But the values (3, 8) and (-3, -1) can satisfy other conditions such as symmetry and being opposite vertices of a rectangle.

Therefore, the correct answers are:

• (A): (3, 8)
• (C): (-3, -1)