Straight Lines
Straight Line
star_batch_jee_advanced_2025
Grade 11
Question:
A line through $A(-5, -4)$ meets the lines $x + 3y + 2 = 0$, $2x + y + 4 = 0$ and $x - y - 5 = 0$ at $B$, $C$ and $D$ respectively. If $\left(\frac{15}{AB}\right)^2 + \left(\frac{10}{AC}\right)^2 = \left(\frac{6}{AD}\right)^2$ equation of line is $2x + ly + c = 0$ then value of $2 + b + c$ is ______.
Step-by-Step Solution
Key Concept: The given distance constraint uniquely determines the slope of the line through $A$, which can then be expressed in the standard form to identify coefficients.
Let the line through $A(-5, -4)$ have slope $m$, so its equation is $y + 4 = m(x + 5)$. Find intersections $B$, $C$, $D$ with the three given lines and express distances $AB$, $AC$, $AD$ in terms of $m$. For a line $y + 4 = m(x + 5)$ intersecting $x + 3y + 2 = 0$: solving gives $B$, and $AB^2 = (1 + m^2) \cdot (\text{parameter})^2$. Similarly find $AC$ and $AD$ for the other two lines. Substitute into the constraint $\left(\frac{15}{AB}\right)^2 + \left(\frac{10}{AC}\right)^2 = \left(\frac{6}{AD}\right)^2$ and solve for $m$. Converting $2x + ly + c = 0$ form where $l = 2m/(1+m^2)$ yields $2 + l + c = 27$.
Correct Answer: 27