If the orthocentre of the triangle formed by the lines y = x + 1$y = x + 1$, y = 4x - 8$y = 4x - 8$ and y = mx + c$y = mx + c$ is at (3, -1)$(3, -1)$, then m - c$m - c$ is:

### Related Formula The product of slopes of two mutually perpendicular lines is always equal to -1$-1$: m_1 cdot m_2 = -1$$m_1 \cdot m_2 = -1$$ ### Core Logic Let the vertices of the triangle be P$P$, Q$Q$, and R$R$. The lines given are: 1) y = x + 1$y = x + 1$ 2) y = 4x - 8$y = 4x - 8$ 3) y = mx + c$y = mx + c$ Solving lines y = x + 1$y = x + 1$ and y = 4x - 8$y = 4x - 8$ gives the vertex P(3, 4)$P(3, 4)$. The orthocentre is given as H(3, -1)$H(3, -1)$. Notice that the x$x$-coordinate of P$P$ and H$H$ are identical (x = 3$x = 3$). This implies that the altitude from vertex P$P$ to the base line y = mx + c$y = mx + c$ is a vertical line along x = 3$x = 3$.

### Step 1: Determine the Slopes Since the altitude from P$P$ is vertical, the side opposite to it (which lies on y = mx + c$y = mx + c$) must be a horizontal line. Therefore, the slope of the line y = mx + c$y = mx + c$ must be zero: m = 0$m = 0$ ### Step 2: Solve for c Let's find point Q$Q$ by intersecting y = x + 1$y = x + 1$ and y = mx + c$y = mx + c$. Since m = 0$m = 0$, y = c$y = c$, we get Q(c-1, c)$Q(c-1, c)$. Using the property that the line segment connecting Q$Q$ to the opposite side's altitude is perpendicular to line PR$PR$ (y = 4x - 8$y = 4x - 8$): textSlope of QH cdot textSlope of PR = -1$$\text{Slope of } QH \cdot \text{Slope of } PR = -1$$ frac-1 - c3 - (c - 1) cdot 4 = -1$$\frac{-1 - c}{3 - (c - 1)} \cdot 4 = -1$$ frac-4(c + 1)4 - c = -1 implies 4c + 4 = 4 - c implies 5c = 0 implies c = 0$$\frac{-4(c + 1)}{4 - c} = -1 \implies 4c + 4 = 4 - c \implies 5c = 0 \implies c = 0$$ ### Step 3: Evaluate m - c Substituting the values of m$m$ and c$c$: m - c = 0 - 0 = 0$$m - c = 0 - 0 = 0$$ ### Pattern Recognition When the x$x$-coordinate of a vertex matches the x$x$-coordinate of the orthocentre, the altitude is vertical, forcing the opposite base to be purely horizontal (m=0$m=0$). This observation cuts down calculation time completely. ### Evaluation Rubric / Model Answer null ### Chapter Mix Class 11 Mathematics: Straight Lines