Conic Sections
Conic Section
Allen Star Batch
Grade 11
Question:
From the points $(x_1, y_1)$ and $(x_2, y_2)$ tangents are drawn to the hyperbola $xy = c^2$, such that a circle passes through these points and the four points of contact, then:
$x_1 y_1 = x_2 y_2$
$x_1 x_2 = y_2 y_2$
$x_1 y_2 + x_2 y_1 = 4c^2$
$x_1 y_1 + x_2 y_2 = 4c^2$
Step-by-Step Solution
Key Concept: For a circle to pass through two external points and four contact points of tangents drawn from these points to hyperbola xy = c², the chord of contact equations xy₁ + x₁y = 2c² and xy₂ + x₂y = 2c² must satisfy orthogonality and confocal conic conditions, leading to x₁x₂ = y₁y₂ and x₁y₂ + x₂y₁ = 4c².
Chords of contact from points $P(x_1, y_1)$ and $Q(x_2, y_2)$ to curve $xy = c^2$ are $xy_1 + x_1y = 2c^2$ and $xy_2 + x_2y = 2c^2$ respectively. The conic through intersection points of $xy = c^2$ with these two chords is found using the family equation. Setting $y_1y_2 = x_1x_2$ and $x_1y_2 + x_2y_1 + \lambda = 0$ gives $\lambda = -4c^2$, yielding $x_1y_2 + x_2y_1 = 4c^2$.
Correct Answer: 2,3