Find the orthocenter of the triangle formed by the lines 7x + y – 10

1.	Find the orthocenter of the triangle formed by the lines 7x + y – 10 = 0, x – 2y + 5 = 0, x + y + 2 = 0.
Answer» Given lines are 7x + y – 10 = 0 → (1), x – 2y + 5 = 0 → (2), x + y + 2 = 0 → (3) Point of intersection of (2) and (3) is B (–3, 1). Equation of the altitude through B is (x + 3) – 7(y – 1) = 0 ⇒ x – 7y + 10 = 0 → (4) Point of intersection of (3) and (1) is C(2, –4) Equation of the altitude through C is 2(x – 2) + (y + 4) = 0 ⇒ 2x + y = 0 → (5) Solving (4) and (5), orthocenter is (- 2/3, 4/3).

Find the orthocenter of the triangle formed by the lines 7x + y – 10 = 0, x – 2y + 5 = 0, x + y + 2 = 0.

Answer»

Given lines are 7x + y – 10 = 0 → (1), x – 2y + 5 = 0 → (2), x + y + 2 = 0 → (3)

Point of intersection of (2) and (3) is B (–3, 1).

Equation of the altitude through B is (x + 3) – 7(y – 1) = 0

⇒ x – 7y + 10 = 0 → (4)

Point of intersection of (3) and (1) is C(2, –4)

Equation of the altitude through C is 2(x – 2) + (y + 4) = 0

⇒ 2x + y = 0 → (5)

Solving (4) and (5), orthocenter is (- 2/3, 4/3).

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Find the orthocenter of the triangle formed by the lines 7x + y – 10 = 0, x – 2y + 5 = 0, x + y + 2 = 0.

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