Question
dentify an axis of revolution and generating conic of the surface
4x2 + 25y2 + 4z2 = 100 . Does this conic also generate x2/4 + y2/25 + z2/4 = 1? Give reasons for your answer.
Answer :
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Let's solve this step-by-step: ### Step 1: Analyze the given equation The equation of the surface is: \[ 4x^2 + 25y^2 + 4z^2 = 100 \] This equation describes an ellipsoid. ### Step 2: Rewriting the equation To better understand its form, divide the whole equation by 100: \[ \frac{4x^2}{100} + \frac{25y^2}{100} + \frac{4z^2}{100} = 1 \] Simplifying each term: \[ \frac{x^2}{25} + \frac{y^2}{4} + \frac{z^2}{25} = 1 \] ### Step 3: Conic and Axis of Revolution This equation represents an ellipsoid that is symmetric around the x-axis, y-axis, and z-axis. The semi-principal axes are \( \sqrt{25} = 5 \) along the x-axis, \( \sqrt{4} = 2 \) along the y-axis, and \( \sqrt{25} = 5 \)
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