Straight Lines
Straight Line
star_batch_jee_advanced_2025
Grade 11
Question:
$A(x_1, y_1), B(x_2, y_2), (y_1 < y_2)$ are two points on the line $x + y = 4$ from which perpendicular $AQ$ and $BP$ are drawn on line $4x + 3y = 10$ where $P$ and $Q$ are the feet of perpendicular such that $AQ = BP = 1$. Now considering $AB$ as diameter, a circle is drawn which meets the line $4x + 3y = 10$ at $C$ and $D$ such that $C$ is closer to $P$. Then which of the following statement(s) is correct?
the value of $\frac{y_1 + y_2}{x_1 + x_2}$ is equal to $-3$
the length $PQ$ is equal to $14$
length $QD$ is equal to $5\sqrt{2} - 7$
radius of circle obtained is $5\sqrt{2}$ units
Step-by-Step Solution
Key Concept: Use the absolute value condition to generate two linear equations, then solve the resulting system to find intersection points.
Given $x_1 + y_1 = 4$ and $x_2 + y_2 = 4$, we use the condition $\left|\frac{4x_1 + 3y_1 - 10}{5}\right| = 1$ to get $4x_1 + 3y_1 - 10 = \pm 5$. This yields two cases: $4x_1 + 3y_1 = 15$ or $4x_1 + 3y_1 = 5$. Solving the system of equations gives $(x_1, y_1) = (3, 1)$ and $(x_2, y_2) = (-7, 11)$. The ratio $\frac{y_1 + y_2}{x_1 + x_2} = \frac{12}{-4} = -3$, and we verify that $L_2$ divides $L_1$ in ratio $1:1$. Finally, $R = [-2, 6]$, $AR = 5\sqrt{2}$, $OR = 7$, and $RD = 5\sqrt{2} - 7$.
Correct Answer: 1,2,3,4