Straight Lines
Straight Line
star_batch_jee_advanced_2025
Grade 11
Question:
The ends of the base of an isosceles triangle are at $(2, 0)$ and $(0, 1)$ and the equation of one side is $x = 2$ then the orthocenter of the triangle is:
$\left(\frac{3}{4}, \frac{3}{2}\right)$
$\left(\frac{5}{4}, -1\right)$
$\left(\frac{3}{4}, -1\right)$
$\left(\frac{4}{3}, \frac{7}{12}\right)$
Step-by-Step Solution
Key Concept: Apply the slope formula between two points and solve the resulting rational equation to find the parameter value.
Given the slope of $HD$ is $2$, we use the slope formula $\frac{1-\frac{2}{\lambda}}{\lambda-1} = 2$ to find $\lambda$. Solving this equation: $1 - \frac{2}{\lambda} = 2(\lambda - 1)$ gives $\lambda = \frac{5}{4}$. Therefore, $H = \left(\frac{5}{4}, 1\right)$.
Correct Answer: 2