Straight Lines
Straight Line
star_batch_jee_advanced_2025
Grade 11
Question:
A straight line through the point $A(-2, -3)$ cuts the lines $x + 3y = 9$ and $x + y + 1 = 0$ at $B$ and $C$ respectively. If $AB.AC = 20$, then product of slopes of line is____.
Step-by-Step Solution
Key Concept: Parametrizing points on a line using the angle of inclination and distance parameter converts geometric constraints into algebraic equations in $ an heta$.
Points on the line with slope $ an heta$ through $(-2, -3)$ have form $(-2 + r\cos heta, -3 + r\sin heta)$. For $B$ on line $x + 3y = 9$: $r_1 = rac{20}{\cos heta + 3\sin heta}$; for $C$ on line $x + y + 1 = 0$: $r_2 = rac{4}{\cos heta + \sin heta}$. Given $r_1r_2 = 20$, we derive $(\cos heta + 3\sin heta)(\cos heta + \sin heta) = 4$, which simplifies to $ an^2 heta - 4 an heta + 3 = 0$, giving $ an heta = 1$ or $3$.
Correct Answer: 3