Hyperbola
Pair of Tangents
IIT-JEE 1999
Grade 11
Question:
If $x = 9$ is the chord of contact of the hyperbola $x^2 - y^2 = 9$, then the pair of tangents from the corresponding external point is:
(1) $9x^2 - 8y^2 + 18x - 9 = 0$
(2) $9x^2 - 8y^2 - 18x + 9 = 0$
(3) $9x^2 - 8y^2 - 18x - 9 = 0$
(4) $9x^2 - 8y^2 + 18x + 9 = 0$
Step-by-Step Solution
Key Concept: $x = 9$ is chord of contact of $x^2 - y^2 = 9$ from $P = (h, k)$: $T = hx - ky = 9$. With $x = 9$: $9h - ky = 9$. This is $x = 9$ only if $h = 1, k = 0$. So $P = (1, 0)$. Pair of tangents: $SS_1 = T^2$, i.e., $(x^2 - y^2 - 9)(1 - 9) = (x - 9)^2$. Expand.
The detailed step-by-step mathematical proof is available inside the Mathbee app workspace.
Correct Answer: (2)