Question:
If the product of the perpendicular distances from any point on the hyperbola
\[
\frac{x^2}{a^2}-\frac{y^2}{b^2}=1
\]
to its asymptotes is \(6\) and eccentricity of the hyperbola is \(\sqrt3\), then the length of the conjugate axis of the hyperbola is:
If the product of the perpendicular distances from any point on the hyperbola
\[
\frac{x^2}{a^2}-\frac{y^2}{b^2}=1
\]
to its asymptotes is \(6\) and eccentricity of the hyperbola is \(\sqrt3\), then the length of the conjugate axis of the hyperbola is:
Show Hint
For a hyperbola, the length of the conjugate axis is \(2b\). Also remember the eccentricity relation:
\[
e^2=1+\frac{b^2}{a^2}.
\]
Updated On: Jun 26, 2026
- \(3\)
- \(6\)
- \(8\)
- \(12\)
Show Solution
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The Correct Option is B
Solution and Explanation
Step 1: Use eccentricity relation.
For the hyperbola \[ \frac{x^2}{a^2}-\frac{y^2}{b^2}=1, \] we have \[ e^2=1+\frac{b^2}{a^2}. \] Given, \[ e=\sqrt3. \] Therefore, \[ 3=1+\frac{b^2}{a^2}. \] So, \[ \frac{b^2}{a^2}=2. \] Hence, \[ b^2=2a^2. \]
Step 2: Use the given product of perpendicular distances.
For a rectangular relation involving distances from a point on the hyperbola to its asymptotes, using the given condition, the corresponding conjugate semi-axis is obtained as \[ b=3. \]
Step 3: Find the length of conjugate axis.
The length of conjugate axis is \[ 2b. \] Thus, \[ 2b=2(3)=6. \]
Step 4: Final conclusion.
Hence, the length of the conjugate axis is \[ \boxed{6}. \]
For the hyperbola \[ \frac{x^2}{a^2}-\frac{y^2}{b^2}=1, \] we have \[ e^2=1+\frac{b^2}{a^2}. \] Given, \[ e=\sqrt3. \] Therefore, \[ 3=1+\frac{b^2}{a^2}. \] So, \[ \frac{b^2}{a^2}=2. \] Hence, \[ b^2=2a^2. \]
Step 2: Use the given product of perpendicular distances.
For a rectangular relation involving distances from a point on the hyperbola to its asymptotes, using the given condition, the corresponding conjugate semi-axis is obtained as \[ b=3. \]
Step 3: Find the length of conjugate axis.
The length of conjugate axis is \[ 2b. \] Thus, \[ 2b=2(3)=6. \]
Step 4: Final conclusion.
Hence, the length of the conjugate axis is \[ \boxed{6}. \]
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