Straight Lines
Straight Line
star_batch_jee_advanced_2025
Grade None
Question:
Consider the triangle $OAB$ where $O = (0,0), B(3,4)$. If orthocenter of triangle is $H(1, 4)$, then coordinates of $A'$ is:
\left(0, \frac{15}{4}\right)
\left(0, \frac{17}{4}\right)
\left(0, \frac{21}{4}\right)
\left(0, \frac{19}{4}\right)
Step-by-Step Solution
Key Concept: Perpendicularity between lines is expressed through negative reciprocal slopes; the foot of perpendicular from a point to an axis lies directly on that axis.
Since $BH \perp OA$ and $OA$ lies on the $y$-axis, line $BH$ is horizontal with equation $y = 4$. The line $AH$ passes through $B(3,4)$ with slope $-\frac{3}{4}$ (negative reciprocal of $\frac{4}{3}$), giving $y - 4 = -\frac{3}{4}(x-1)$. Setting $x = 0$ yields $y = 4 + \frac{3}{4} = \frac{19}{4}$, so $A = (0, \frac{19}{4})$.
Correct Answer: 4