Given the equation, 6𝑥2 + 24𝑦2 = 24, explain how you would determine which of the four conic sectionsit is based on its equation.3. Explain what happens if the equation is changed to 6𝑥2 − 24𝑦2 = 24 .
Question
Given the equation, 6𝑥2 + 24𝑦2 = 24, explain how you would determine which of the four conic sectionsit is based on its equation.3. Explain what happens if the equation is changed to 6𝑥2 − 24𝑦2 = 24 .
Solution
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To determine the type of conic section represented by the equation 6x^2 + 24y^2 = 24, we first need to rewrite the equation in standard form. We can do this by dividing the entire equation by 24 to simplify it. This gives us x^2/4 + y^2/1 = 1.
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The standard form of the equation of a conic section helps us identify the type of conic. The general forms are as follows:
- Circle: x^2 + y^2 = r^2
- Ellipse: x^2/a^2 + y^2/b^2 = 1 (a ≠ b)
- Hyperbola: x^2/a^2 - y^2/b^2 = 1 or y^2/b^2 - x^2/a^2 = 1
- Parabola: y^2 = 4ax or x^2 = 4ay
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Looking at our simplified equation, x^2/4 + y^2/1 = 1, we can see that it matches the standard form of an ellipse (x^2/a^2 + y^2/b^2 = 1), where a^2 = 4 and b^2 = 1. Therefore, the given equation represents an ellipse.
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If the equation is changed to 6x^2 - 24y^2 = 24, we again start by simplifying it to standard form by dividing by 24. This gives us x^2/4 - y^2/1 = 1.
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Comparing this to the standard forms of conic sections, we can see that it matches the form of a hyperbola (x^2/a^2 - y^2/b^2 = 1), where a^2 = 4 and b^2 = 1. Therefore, the modified equation represents a hyperbola.
Similar Questions
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What are the coordinates of the center of the conic section with equation 4x² + 9y ² − 24x + 36y + 36 = 0?1 pointA. (−3, 2)B. (−2, 3)C. (3, −2)D. (2, −3)
How can you distinguish parabolas from other conic sections by looking at their equations without an 𝑥𝑦−term?Group of answer choicesIn equations for parabolas, 𝑥 and 𝑦 are both squared and the coefficients on them are the same, including the sign.In equations for parabolas, 𝑥 and 𝑦 are both squared, and exactly one of the coefficients is negative and exactly one of the coefficients is positive.In equations for parabolas, 𝑥 and 𝑦 are both squared, and the coefficients are positive but different.In equations for parabolas, either 𝑥 and 𝑦 is squared, but not both.
Which type of conic section does the following equation represent?left bracket, x, minus, 7, right bracket, squared, equals, minus, 28, left bracket, y, plus, 6, right bracket(x−7) 2 =−28(y+6)
Conic Sections are provided here to enhance the Maths exam preparation of the students and to help them score better marks. Our subject matter experts have developed these NCERT Solutions for Chapter 11 in accordance with the updated syllabus of CBSE Class 11 for 2023-24. Moreover, the solutions have been designed to match the understanding level of the students, so that they can grasp the difficult Maths concepts with ease.
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