Maths Q11 – JEE-MAIN 2025

If the orthocenter of the triangle formed by the lines

y = x + 1

y = 4 x - 8

and

y = m x + c

is at

(3, - 1)

, then

m - c

is :

Clue 1 of 3: Orthocenter Definition

The orthocenter is the intersection point of the altitudes of a triangle. An altitude from a vertex is perpendicular to the opposite side.

🥷

Ninja Strategy•Vertical Altitude Insight

Notice that vertex A (intersection of $L_{1}$ and $L_{2}$ ) and the orthocenter H share the same x-coordinate. This implies the altitude from A is a vertical line, which immediately determines the slope of the opposite side $L_{3}$ to be 0.

🧠 Full Solution Path

Step 1: Determine the slope of the third line (

m

).✦ Active

Let the three lines be $L_{1} : y = x + 1$ , $L_{2} : y = 4 x - 8$ , and $L_{3} : y = m x + c$ . The orthocenter is $H (3, - 1)$ . First, find the intersection point of $L_{1}$ and $L_{2}$ (let's call it vertex $A$ ):

x + 1 = 4 x - 8 ⟹ 3 x = 9 ⟹ x = 3

Substitute $x = 3$ into $y = x + 1$ to get $y = 3 + 1 = 4$ . So, vertex $A$ is $(3, 4)$ . The altitude from vertex $A$ to the opposite side $L_{3}$ must pass through the orthocenter $H (3, - 1)$ . Since $A (3, 4)$ and $H (3, - 1)$ have the same x-coordinate, the line $A H$ is a vertical line ( $x = 3$ ). For $A H$ to be an altitude, it must be perpendicular to $L_{3}$ . A vertical line is perpendicular to a horizontal line. Therefore, $L_{3}$ must be a horizontal line, which means its slope $m$ must be $0$ . So, $L_{3}$ is $y = c$ .

Step 2: Determine the y-intercept of the third line (

c

).○ Expand

Now consider the altitude from vertex $B$ (intersection of $L_{1}$ and $L_{3}$ ) to side $A C$ ( $L_{2}$ ). The slope of $L_{2} : y = 4 x - 8$ is $m_{2} = 4$ . The slope of the altitude perpendicular to $L_{2}$ is $- \frac{1}{4}$ . This altitude passes through the orthocenter $H (3, - 1)$ . Its equation is:

y - (- 1) = - \frac{1}{4} (x - 3) ⟹ 4 (y + 1) = - (x - 3) ⟹ 4 y + 4 = - x + 3 ⟹ x + 4 y + 1 = 0

Vertex $B$ is the intersection of $L_{1} : y = x + 1$ and $L_{3} : y = c$ (since $m = 0$ ). Substitute $y = c$ into $y = x + 1$ : $c = x + 1 ⟹ x = c - 1$ . So, vertex $B$ is $(c - 1, c)$ . Since $B$ lies on the altitude $x + 4 y + 1 = 0$ , substitute its coordinates into the altitude equation:

(c - 1) + 4 (c) + 1 = 0 ⟹ c - 1 + 4 c + 1 = 0 ⟹ 5 c = 0 ⟹ c = 0

💡 Teacher's Secret Hint

Remember that all three altitudes must pass through the orthocenter. Using any two altitudes is sufficient to find the unknown parameters.

Step 3: Calculate

m - c

.○ Expand

We found $m = 0$ and $c = 0$ . Therefore, the value of $m - c$ is:

m - c = 0 - 0 = 0

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Maths Question 11 – JEE-MAIN 2025

Clue 1 of 3: Orthocenter Definition

🧠 Full Solution Path