Intersection of Two Lines

Step 1: Find the Vertices of the Triangle
By solving the line equations in pairs, we find the three intersection points (vertices):
- Vertex A (Intersection of $$x+y=7$$ and $$3x+4y=25$$): $$(3, 4)$$
- Vertex B (Intersection of $$3x+4y=25$$ and $$x+2y=10$$): $$(5, 2.5)$$
- Vertex C (Intersection of $$x+y=7$$ and $$x+2y=10$$): $$(4, 3)$$

Step 2: Find the Equations of Two Altitudes
The orthocentre is the intersection point of the altitudes.

- Altitude from Vertex A $$(3,4)$$ to side BC ($$x+2y=10$$):
- The slope of side BC is $$-\frac{1}{2}$$.
- The slope of the perpendicular altitude is $$2$$.
- Equation: $$y - 4 = 2(x - 3)$$ which simplifies to $$2x - y = 2$$

- Altitude from Vertex C $$(4,3)$$ to side AB ($$3x+4y=25$$):
- The slope of side AB is $$-\frac{3}{4}$$.
- The slope of the perpendicular altitude is $$\frac{4}{3}$$.
- Equation: $$y - 3 = \frac{4}{3}(x - 4)$$ which simplifies to $$4x - 3y = 7$$

Step 3: Find the Intersection of the Altitudes
Solve the system of equations from the two altitudes:
1. $$y = 2x - 2$$
2. $$4x - 3y = 7$$

Substitute equation (1) into equation (2):
$$4x - 3(2x - 2) = 7$$
$$4x - 6x + 6 = 7$$
$$-2x = 1$$
$$x = -0.5$$

Substitute $$x = -0.5$$ back into equation (1):
$$y = 2(-0.5) - 2$$
$$y = -1 - 2$$
$$y = -3$$

Therefore, the coordinates of the orthocentre are $$(-0.5, -3)$$.

1. Find the fixed point of intersection $$P$$:

We solve the linear system of equations to isolate our fixed pivot point:

$$x + 2y = 3$$

$$2x - y = 1 \implies y = 2x - 1$$

Substitute the expression for $$y$$ directly into the first equation:

$$x + 2(2x - 1) = 3$$

$$x + 4x - 2 = 3$$

$$5x = 5 \implies x = 1$$

$$y = 2(1) - 1 = 1$$

So, the fixed point of intersection is $$P(1, 1)$$.

2. Apply the maximum distance geometric principle:

For a family of lines pivoting through a fixed point $$P$$, the perpendicular distance from the origin $$O(0,0)$$ to the line reaches its absolute maximum length when the line itself is drawn perfectly perpendicular to the position segment line $$OP$$.

The slope of the straight segment line $$OP$$ connecting $$(0,0)$$ and $$(1,1)$$ is:

$$m_{OP} = \frac{1 - 0}{1 - 0} = 1$$

Since our target variable line must stand perpendicular to $$OP$$, its matching slope $$m$$ is:

$$m \times m_{OP} = -1 \implies m = -1$$

3. Formulate the equation of the line:

The line has a slope of $$-1$$ and passes through the coordinate point $$P(1,1)$$. Using the point-slope formula:

$$y - 1 = -1(x - 1)$$

$$y - 1 = -x + 1$$

$$x + y - 2 = 0$$

4. Compare with the standard form and evaluate:

Comparing $$x + y - 2 = 0$$ to the template $$ax + by + c = 0$$, we extract the parameters:

$$a = 1$$

$$b = 1$$

$$c = -2$$

Substitute these coefficient values into our target expression:

$$a^2 + b^2 - c^2 = (1)^2 + (1)^2 - (-2)^2 = 1 + 1 - 4 = -2$$

We are given the system of linear equations:

1. $$3x + 4y = 9$$
2. $$y = mx + 1$$

Substitute the expression for $$y$$ from equation (2) into equation (1):

$$3x + 4(mx + 1) = 9$$

Expand the brackets:

$$3x + 4mx + 4 = 9$$

Combine the terms containing $$x$$ and move the constant term to the right-hand side:

$$(3 + 4m)x = 9 - 4$$  $$(3 + 4m)x = 5$$

Solving for $$x$$, we get:

$$x = \frac{5}{3 + 4m}$$

For the $$x$$-coordinate to be an integer, the denominator $$(3 + 4m)$$ must be an integer factor of the numerator $$5$$.

The integer factors of $$5$$ are:

$$\pm 1, \pm 5$$

We can set up four separate cases to test which of these factors result in an integer value for $$m$$:

• Case 1: $$3 + 4m = 1$$ $$4m = 1 - 3 \implies 4m = -2 \implies m = -\frac{1}{2} \quad (\text{Not an integer})$$
• Case 2: $$3 + 4m = -1$$ $$4m = -1 - 3 \implies 4m = -4 \implies m = -1 \quad (\mathbf{Valid \text{ } Integer})$$
• Case 3: $$3 + 4m = 5$$ $$4m = 5 - 3 \implies 4m = 2 \implies m = \frac{1}{2} \quad (\text{Not an integer})$$
• Case 4: $$3 + 4m = -5$$ $$4m = -5 - 3 \implies 4m = -8 \implies m = -2 \quad (\mathbf{Valid \text{ } Integer})$$

There are exactly two integer values of $$m$$ ($$m = -1$$ and $$m = -2$$) that ensure the $$x$$-coordinate of the intersection point is an integer.

• When $$m = -1$$, $$x = \frac{5}{3 - 4} = -5$$
• When $$m = -2$$, $$x = \frac{5}{3 - 8} = -1$$

Thus, the number of integral values of $$m$$ is $$2$$, matching Option A.