Circles
Circle
star_batch_jee_advanced_2025
Grade None
Question:
Let $L_1$ be a straight line passing through the origin and $L_2$ be the straight line $x + y = 1$. If the intercepts made by the circle $x^2 + y^2 - x + 3y = 0$ on $L_1$ and $L_2$ are equal, which of the following equations can represent $L_1$.
$x + y = 0
$x - y = 0
$x + 7y = 0
$x - 7y = 0
Step-by-Step Solution
Key Concept: Equal chords of a circle are equidistant from the center; use this to find lines through the origin whose chords match the given line's chord length.
The circle is $x^2 + y^2 - x + 3y = 0$ with center $H = (\frac{1}{2}, -\frac{3}{2})$. The given line is $x + y - 1 = 0$. For any line $y = mx$ through the origin, equal chords are equidistant from $H$. Setting the distances equal: $\frac{|\frac{1}{2} - \frac{3}{2}|}{\sqrt{2}} = \frac{|\frac{3}{2} - m \cdot \frac{1}{2}|}{\sqrt{1+m^2}}$ simplifies to $7m^2 - 6m - 1 = 0$, giving $m = 1$ or $m = -\frac{1}{7}$. The equations are $x - y = 0$ or $x + 7y = 0$.
Correct Answer: 2,3