Straight Lines
Orthocentre and quadratic equation from its coordinates
nta_pyq_2023_jan
Grade 11
Question:
If the orthocentre of the triangle, whose vertices are $(1, 2)$, $(2, 3)$ and $(3, 1)$ is $(\alpha, \beta)$, then the quadratic equation whose roots are $\alpha + 4\beta$ and $4\alpha + \beta$, is
x^2 - 19x + 90 = 0
x^2 - 18x + 80 = 0
x^2 - 22x + 120 = 0
x^2 - 20x + 99 = 0
Step-by-Step Solution
Key Concept: Find the orthocentre by setting up two altitude equations (each altitude is perpendicular to a side and passes through the opposite vertex) and solving.
$\alpha = \frac{5}{3}$, $\beta = \frac{7}{3}$. $\alpha + 4\beta = \frac{5}{3} + \frac{28}{3} = 11$. $4\alpha + \beta = \frac{20}{3} + \frac{7}{3} = 9$. Sum $= 20$, product $= 99$. Equation: $x^2 - 20x + 99 = 0$.
Correct Answer: 4