Explore Westonci.ca, the premier Q&A site that helps you find precise answers to your questions, no matter the topic. Connect with a community of experts ready to help you find solutions to your questions quickly and accurately. Get precise and detailed answers to your questions from a knowledgeable community of experts on our Q&A platform.
Sagot :
To find the equation of the locus of a point such that the sum of its distances from the points \((0,2)\) and \((0,-2)\) is 6 units, we begin by noting that such a locus describes an ellipse. Here’s a structured step-by-step solution:
1. Identify the Foci and Sum of Distances:
- The points \((0, 2)\) and \((0, -2)\) are the foci of the ellipse.
- The sum of distances from any point on the ellipse to these two fixed points (foci) is given as 6 units.
2. Determine the Semi-Major Axis (a):
- In an ellipse, the sum of distances from any point on the ellipse to the foci is equal to the length of the major axis, which is \(2a\).
- Given sum of distances is \(6\) units, so \(2a = 6\).
- Thus, the semi-major axis, \(a\), is \(a = \frac{6}{2} = 3\).
3. Calculate the Distance Between the Foci (2c):
- The distance between the foci \((0, 2)\) and \((0, -2)\) is calculated as \(2c\).
- The distance between the points \((0, 2)\) and \((0, -2)\) is \(4\) units.
- Hence, \(2c = 4\) which gives \(c = 2\).
4. Determine the Semi-Minor Axis (b):
- Using the relationship for ellipses: \(c^2 = a^2 - b^2\), we can solve for \(b\).
- Here, we have \(a = 3\) and \(c = 2\).
- Substitute these values into the equation:
[tex]\[ c^2 = a^2 - b^2 \][/tex]
[tex]\[ 2^2 = 3^2 - b^2 \][/tex]
[tex]\[ 4 = 9 - b^2 \][/tex]
[tex]\[ b^2 = 9 - 4 \][/tex]
[tex]\[ b^2 = 5 \][/tex]
- Therefore, \(b = \sqrt{5} \approx 2.236\).
5. Formulate the Equation of the Ellipse:
- The standard form of the equation of an ellipse centered at the origin \((0,0)\) with semi-major axis \(a\) and semi-minor axis \(b\) is:
[tex]\[ \frac{x^2}{a^2} + \frac{y^2}{b^2} = 1 \][/tex]
- Plugging in the values of \(a\) and \(b\):
[tex]\[ \frac{x^2}{3^2} + \frac{y^2}{(\sqrt{5})^2} = 1 \][/tex]
[tex]\[ \frac{x^2}{9} + \frac{y^2}{5} = 1 \][/tex]
Therefore, the equation of the locus of a point such that the sum of its distances from \((0,2)\) and \((0,-2)\) is 6 units is:
[tex]\[ \boxed{\frac{x^2}{9} + \frac{y^2}{5} = 1} \][/tex]
1. Identify the Foci and Sum of Distances:
- The points \((0, 2)\) and \((0, -2)\) are the foci of the ellipse.
- The sum of distances from any point on the ellipse to these two fixed points (foci) is given as 6 units.
2. Determine the Semi-Major Axis (a):
- In an ellipse, the sum of distances from any point on the ellipse to the foci is equal to the length of the major axis, which is \(2a\).
- Given sum of distances is \(6\) units, so \(2a = 6\).
- Thus, the semi-major axis, \(a\), is \(a = \frac{6}{2} = 3\).
3. Calculate the Distance Between the Foci (2c):
- The distance between the foci \((0, 2)\) and \((0, -2)\) is calculated as \(2c\).
- The distance between the points \((0, 2)\) and \((0, -2)\) is \(4\) units.
- Hence, \(2c = 4\) which gives \(c = 2\).
4. Determine the Semi-Minor Axis (b):
- Using the relationship for ellipses: \(c^2 = a^2 - b^2\), we can solve for \(b\).
- Here, we have \(a = 3\) and \(c = 2\).
- Substitute these values into the equation:
[tex]\[ c^2 = a^2 - b^2 \][/tex]
[tex]\[ 2^2 = 3^2 - b^2 \][/tex]
[tex]\[ 4 = 9 - b^2 \][/tex]
[tex]\[ b^2 = 9 - 4 \][/tex]
[tex]\[ b^2 = 5 \][/tex]
- Therefore, \(b = \sqrt{5} \approx 2.236\).
5. Formulate the Equation of the Ellipse:
- The standard form of the equation of an ellipse centered at the origin \((0,0)\) with semi-major axis \(a\) and semi-minor axis \(b\) is:
[tex]\[ \frac{x^2}{a^2} + \frac{y^2}{b^2} = 1 \][/tex]
- Plugging in the values of \(a\) and \(b\):
[tex]\[ \frac{x^2}{3^2} + \frac{y^2}{(\sqrt{5})^2} = 1 \][/tex]
[tex]\[ \frac{x^2}{9} + \frac{y^2}{5} = 1 \][/tex]
Therefore, the equation of the locus of a point such that the sum of its distances from \((0,2)\) and \((0,-2)\) is 6 units is:
[tex]\[ \boxed{\frac{x^2}{9} + \frac{y^2}{5} = 1} \][/tex]
We appreciate your time on our site. Don't hesitate to return whenever you have more questions or need further clarification. Thanks for using our service. We're always here to provide accurate and up-to-date answers to all your queries. Stay curious and keep coming back to Westonci.ca for answers to all your burning questions.
