Hyperbola
Locus from Chord of Contact
Premium Question
Grade 11
Question:
Tangents are drawn from any point on the hyperbola $\frac{x^2}{9} - \frac{y^2}{4} = 1$ to the circle $x^2 + y^2 = 9$. The locus of the midpoint of the chord of contact is:
(1) $\frac{x^2}{9} - \frac{y^2}{4} = \frac{(x^2 + y^2)^2}{81}$
(2) $\frac{x^2}{9} + \frac{y^2}{4} = \frac{(x^2 + y^2)^2}{9}$
(3) $x^2 + y^2 = 9$
(4) $(x^2 + y^2)^2 = 9(x^2 - y^2)$
Step-by-Step Solution
Key Concept: Let $P = (h, k)$ be on hyperbola; chord of contact to $x^2 + y^2 = 9$ from $P$: $hx + ky = 9$. Midpoint of chord $(x_0, y_0)$: from $T = S_1$ for the circle, $x_0x + y_0y = x_0^2 + y_0^2$. Equating with $hx + ky = 9$: $h/x_0 = k/y_0 = 9/(x_0^2 + y_0^2)$, so $h = 9x_0/(x_0^2 + y_0^2)$, $k = 9y_0/(x_0^2 + y_0^2)$. Substituting into hyperbola equation.
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Correct Answer: (1)