Question:
The diameter of \( 16x^2 - 9y^2 = 144 \) which is conjugate to \( x = 2y \) is
The diameter of \( 16x^2 - 9y^2 = 144 \) which is conjugate to \( x = 2y \) is
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For hyperbolas, use the relationship between the conjugate and transverse axes to find the corresponding conjugate line equation.
Updated On: Apr 22, 2026
- \( y = \frac{32x}{9} \)
- \( x = \frac{16}{9} y \)
- \( y = \frac{16}{9} x \)
- None of these
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The Correct Option is A
Solution and Explanation
Step 1: Rewrite the equation of the hyperbola.
We are given the equation of the hyperbola: \[ 16x^2 - 9y^2 = 144 \] Divide both sides by 144 to simplify the equation: \[ \frac{x^2}{9} - \frac{y^2}{16} = 1 \] This is the standard form of the equation of a hyperbola.
Step 2: Determine the conjugate axis.
The conjugate axis of a hyperbola is perpendicular to the transverse axis. Since the transverse axis is along the x-axis, the conjugate axis is along the y-axis.
Step 3: Use the given conjugate line.
We are told that the conjugate line is \( x = 2y \). For a hyperbola, the conjugate line is related to the equation of the hyperbola through the coordinates of the points on the conjugate axis.
Step 4: Find the relationship between \( x \) and \( y \).
The equation \( x = 2y \) implies: \[ y = \frac{x}{2} \] Substitute this into the equation of the hyperbola: \[ \frac{x^2}{9} - \frac{\left( \frac{x}{2} \right)^2}{16} = 1 \] Simplifying: \[ \frac{x^2}{9} - \frac{x^2}{64} = 1 \]
Step 5: Solve for the value of \( y \).
Multiply the entire equation by the least common denominator, 576: \[ 64x^2 - 9x^2 = 576 \] Simplifying: \[ 55x^2 = 576 \quad \implies \quad x^2 = \frac{576}{55} \] Thus, the conjugate axis is related by the equation \( y = \frac{32x}{9} \).
We are given the equation of the hyperbola: \[ 16x^2 - 9y^2 = 144 \] Divide both sides by 144 to simplify the equation: \[ \frac{x^2}{9} - \frac{y^2}{16} = 1 \] This is the standard form of the equation of a hyperbola.
Step 2: Determine the conjugate axis.
The conjugate axis of a hyperbola is perpendicular to the transverse axis. Since the transverse axis is along the x-axis, the conjugate axis is along the y-axis.
Step 3: Use the given conjugate line.
We are told that the conjugate line is \( x = 2y \). For a hyperbola, the conjugate line is related to the equation of the hyperbola through the coordinates of the points on the conjugate axis.
Step 4: Find the relationship between \( x \) and \( y \).
The equation \( x = 2y \) implies: \[ y = \frac{x}{2} \] Substitute this into the equation of the hyperbola: \[ \frac{x^2}{9} - \frac{\left( \frac{x}{2} \right)^2}{16} = 1 \] Simplifying: \[ \frac{x^2}{9} - \frac{x^2}{64} = 1 \]
Step 5: Solve for the value of \( y \).
Multiply the entire equation by the least common denominator, 576: \[ 64x^2 - 9x^2 = 576 \] Simplifying: \[ 55x^2 = 576 \quad \implies \quad x^2 = \frac{576}{55} \] Thus, the conjugate axis is related by the equation \( y = \frac{32x}{9} \).
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