Straight Lines
Straight Line
star_batch_jee_advanced_2025
Grade 11
Question:
Line $\frac{x}{a} + \frac{y}{b} = 1$ cuts the coordinate axes at $A(a, 0)$ and $B(0, b)$ and the line $\frac{x}{a'} + \frac{y}{b'} = -1$ at $A'(-a', 0)$ and $B'(0, -b')$. If the points $A, B, A', B'$ are concyclic then the orthocenter of the triangle $ABA'$ is:
(0, 0)
(0, b')
(0, \frac{bb'}{a})
(0, \frac{bb'}{a})
Step-by-Step Solution
Key Concept: The image of a point under reflection through a circle's center is found using the power of a point theorem.
Since $H$ is the image of $B'(0,b)$ under reflection through circle center $O$, we have $H = (0,b')$. Using the power of point $O$ with respect to the circle, $aa' = bb' = b \cdot b'$, which gives $b' = \frac{aa'}{b}$. Thus $H = (0, \frac{aa'}{b})$.
Correct Answer: 2,3