Question

If \((1,2)\) is the focus, \(x+2y=0\) is the directrix and \(\sqrt{2}\) is the eccentricity of a hyperbola, then the equation of the hyperbola is

• A. \(x^2-y^2=a^2\)
• B. \(3x^2-8xy-3y^2-10x-20y+25=0\)
• C. \(xy=c^2\)
• D. \(3x^2-8xy-3y^2+10x-20y-25=0\)

Hint:

For a conic defined by a focus, directrix, and eccentricity, \[ \frac{\text{Distance from focus}} {\text{Distance from directrix}} =e. \] Squaring the resulting equation usually leads directly to the Cartesian equation of the conic.

Answer 1

The Correct Option is B

Step 1: Apply the focus-directrix definition.
For any point \(P(x,y)\): \(\dfrac{PS}{PM} = e = \sqrt{2}\), where \(S=(1,2)\) and directrix \(x+2y=0\).
\[PS^2 = 2 \cdot PM^2\] \[(x-1)^2+(y-2)^2 = 2\cdot\frac{(x+2y)^2}{5}\]

Step 2: Expand and simplify.
\[5[(x-1)^2+(y-2)^2] = 2(x+2y)^2\] \[5(x^2-2x+1+y^2-4y+4) = 2(x^2+4xy+4y^2)\] \[5x^2-10x+25+5y^2-20y = 2x^2+8xy+8y^2\] \[3x^2-8xy-3y^2-10x-20y+25=0\]
\[ \boxed{3x^2-8xy-3y^2-10x-20y+25=0} \]