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A park is elliptical in shape with semi major axis is 3 units and semi minor axis is 2 units. A partition (straight line) is drawn in park as shown below to make a butterfly park.The points satisfying the equation of straight line(partition) are (0,2) and (3,0). 1. Find the equation of a elliptical park. 2. Find the equation of partition(straight line). 3. Find the total area of the elliptical park. 4.Find area of triangle AOB. 5. Find the area of the butterfly park.?
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A park is elliptical in shape with semi major axis is 3 units and semi...
1. Equation of the elliptical park:
An ellipse can be represented by the equation:
$\frac{(x-h)^2}{a^2} + \frac{(y-k)^2}{b^2} = 1$
where (h, k) is the center of the ellipse, and a and b are the lengths of the semi-major and semi-minor axes, respectively.
Given that the semi-major axis is 3 units and the semi-minor axis is 2 units, we can plug these values into the equation to find the equation of the elliptical park:
$\frac{(x-h)^2}{3^2} + \frac{(y-k)^2}{2^2} = 1$
2. Equation of the partition (straight line):
To find the equation of the partition, we can use the slope-intercept form of a straight line:
$y = mx + c$
where m is the slope of the line and c is the y-intercept.
We are given two points on the line: (0,2) and (3,0). We can use these points to find the slope:
$m = \frac{y_2 - y_1}{x_2 - x_1} = \frac{0 - 2}{3 - 0} = -\frac{2}{3}$
Now we can substitute the slope and one of the points into the slope-intercept form to find the equation of the partition:
$y = -\frac{2}{3}x + c$
Using the point (0,2):
$2 = -\frac{2}{3}(0) + c$
$c = 2$
So the equation of the partition is:
$y = -\frac{2}{3}x + 2$
3. Total area of the elliptical park:
The formula to calculate the area of an ellipse is:
$Area = \pi \cdot a \cdot b$
Substituting the values of a = 3 and b = 2, we get:
$Area = \pi \cdot 3 \cdot 2 = 6\pi$
Therefore, the total area of the elliptical park is 6π square units.
4. Area of triangle AOB:
Triangle AOB is formed by the x-axis, the partition line, and the y-axis. Since the x-axis and the y-axis are perpendicular to each other, the triangle is a right-angled triangle.
The length of the base of the triangle is 3 units (x-coordinate of point B). The height of the triangle is 2 units (y-coordinate of point A).
The area of a right-angled triangle can be calculated using the formula:
$Area = \frac{1}{2} \cdot \text{base} \cdot \text{height}$
Substituting the values, we get:
$Area = \frac{1}{2} \cdot 3 \cdot 2 = 3$
Therefore, the area of triangle AOB is 3 square units.
5. Area of the butterfly park:
The butterfly park is formed by the two halves of the elliptical park separated by the partition line. To find the area of the butterfly park, we need to subtract the area of triangle AOB from the total area of
An ellipse can be represented by the equation:
$\frac{(x-h)^2}{a^2} + \frac{(y-k)^2}{b^2} = 1$
where (h, k) is the center of the ellipse, and a and b are the lengths of the semi-major and semi-minor axes, respectively.
Given that the semi-major axis is 3 units and the semi-minor axis is 2 units, we can plug these values into the equation to find the equation of the elliptical park:
$\frac{(x-h)^2}{3^2} + \frac{(y-k)^2}{2^2} = 1$
2. Equation of the partition (straight line):
To find the equation of the partition, we can use the slope-intercept form of a straight line:
$y = mx + c$
where m is the slope of the line and c is the y-intercept.
We are given two points on the line: (0,2) and (3,0). We can use these points to find the slope:
$m = \frac{y_2 - y_1}{x_2 - x_1} = \frac{0 - 2}{3 - 0} = -\frac{2}{3}$
Now we can substitute the slope and one of the points into the slope-intercept form to find the equation of the partition:
$y = -\frac{2}{3}x + c$
Using the point (0,2):
$2 = -\frac{2}{3}(0) + c$
$c = 2$
So the equation of the partition is:
$y = -\frac{2}{3}x + 2$
3. Total area of the elliptical park:
The formula to calculate the area of an ellipse is:
$Area = \pi \cdot a \cdot b$
Substituting the values of a = 3 and b = 2, we get:
$Area = \pi \cdot 3 \cdot 2 = 6\pi$
Therefore, the total area of the elliptical park is 6π square units.
4. Area of triangle AOB:
Triangle AOB is formed by the x-axis, the partition line, and the y-axis. Since the x-axis and the y-axis are perpendicular to each other, the triangle is a right-angled triangle.
The length of the base of the triangle is 3 units (x-coordinate of point B). The height of the triangle is 2 units (y-coordinate of point A).
The area of a right-angled triangle can be calculated using the formula:
$Area = \frac{1}{2} \cdot \text{base} \cdot \text{height}$
Substituting the values, we get:
$Area = \frac{1}{2} \cdot 3 \cdot 2 = 3$
Therefore, the area of triangle AOB is 3 square units.
5. Area of the butterfly park:
The butterfly park is formed by the two halves of the elliptical park separated by the partition line. To find the area of the butterfly park, we need to subtract the area of triangle AOB from the total area of
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A park is elliptical in shape with semi major axis is 3 units and semi minor axis is 2 units. A partition (straight line) is drawn in park as shown below to make a butterfly park.The points satisfying the equation of straight line(partition) are (0,2) and (3,0). 1. Find the equation of a elliptical park. 2. Find the equation of partition(straight line). 3. Find the total area of the elliptical park. 4.Find area of triangle AOB. 5. Find the area of the butterfly park.?
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A park is elliptical in shape with semi major axis is 3 units and semi minor axis is 2 units. A partition (straight line) is drawn in park as shown below to make a butterfly park.The points satisfying the equation of straight line(partition) are (0,2) and (3,0). 1. Find the equation of a elliptical park. 2. Find the equation of partition(straight line). 3. Find the total area of the elliptical park. 4.Find area of triangle AOB. 5. Find the area of the butterfly park.? for Commerce 2024 is part of Commerce preparation. The Question and answers have been prepared according to the Commerce exam syllabus. Information about A park is elliptical in shape with semi major axis is 3 units and semi minor axis is 2 units. A partition (straight line) is drawn in park as shown below to make a butterfly park.The points satisfying the equation of straight line(partition) are (0,2) and (3,0). 1. Find the equation of a elliptical park. 2. Find the equation of partition(straight line). 3. Find the total area of the elliptical park. 4.Find area of triangle AOB. 5. Find the area of the butterfly park.? covers all topics & solutions for Commerce 2024 Exam. Find important definitions, questions, meanings, examples, exercises and tests below for A park is elliptical in shape with semi major axis is 3 units and semi minor axis is 2 units. A partition (straight line) is drawn in park as shown below to make a butterfly park.The points satisfying the equation of straight line(partition) are (0,2) and (3,0). 1. Find the equation of a elliptical park. 2. Find the equation of partition(straight line). 3. Find the total area of the elliptical park. 4.Find area of triangle AOB. 5. Find the area of the butterfly park.?.
A park is elliptical in shape with semi major axis is 3 units and semi minor axis is 2 units. A partition (straight line) is drawn in park as shown below to make a butterfly park.The points satisfying the equation of straight line(partition) are (0,2) and (3,0). 1. Find the equation of a elliptical park. 2. Find the equation of partition(straight line). 3. Find the total area of the elliptical park. 4.Find area of triangle AOB. 5. Find the area of the butterfly park.? for Commerce 2024 is part of Commerce preparation. The Question and answers have been prepared according to the Commerce exam syllabus. Information about A park is elliptical in shape with semi major axis is 3 units and semi minor axis is 2 units. A partition (straight line) is drawn in park as shown below to make a butterfly park.The points satisfying the equation of straight line(partition) are (0,2) and (3,0). 1. Find the equation of a elliptical park. 2. Find the equation of partition(straight line). 3. Find the total area of the elliptical park. 4.Find area of triangle AOB. 5. Find the area of the butterfly park.? covers all topics & solutions for Commerce 2024 Exam. Find important definitions, questions, meanings, examples, exercises and tests below for A park is elliptical in shape with semi major axis is 3 units and semi minor axis is 2 units. A partition (straight line) is drawn in park as shown below to make a butterfly park.The points satisfying the equation of straight line(partition) are (0,2) and (3,0). 1. Find the equation of a elliptical park. 2. Find the equation of partition(straight line). 3. Find the total area of the elliptical park. 4.Find area of triangle AOB. 5. Find the area of the butterfly park.?.
Solutions for A park is elliptical in shape with semi major axis is 3 units and semi minor axis is 2 units. A partition (straight line) is drawn in park as shown below to make a butterfly park.The points satisfying the equation of straight line(partition) are (0,2) and (3,0). 1. Find the equation of a elliptical park. 2. Find the equation of partition(straight line). 3. Find the total area of the elliptical park. 4.Find area of triangle AOB. 5. Find the area of the butterfly park.? in English & in Hindi are available as part of our courses for Commerce.
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A park is elliptical in shape with semi major axis is 3 units and semi minor axis is 2 units. A partition (straight line) is drawn in park as shown below to make a butterfly park.The points satisfying the equation of straight line(partition) are (0,2) and (3,0). 1. Find the equation of a elliptical park. 2. Find the equation of partition(straight line). 3. Find the total area of the elliptical park. 4.Find area of triangle AOB. 5. Find the area of the butterfly park.?, a detailed solution for A park is elliptical in shape with semi major axis is 3 units and semi minor axis is 2 units. A partition (straight line) is drawn in park as shown below to make a butterfly park.The points satisfying the equation of straight line(partition) are (0,2) and (3,0). 1. Find the equation of a elliptical park. 2. Find the equation of partition(straight line). 3. Find the total area of the elliptical park. 4.Find area of triangle AOB. 5. Find the area of the butterfly park.? has been provided alongside types of A park is elliptical in shape with semi major axis is 3 units and semi minor axis is 2 units. A partition (straight line) is drawn in park as shown below to make a butterfly park.The points satisfying the equation of straight line(partition) are (0,2) and (3,0). 1. Find the equation of a elliptical park. 2. Find the equation of partition(straight line). 3. Find the total area of the elliptical park. 4.Find area of triangle AOB. 5. Find the area of the butterfly park.? theory, EduRev gives you an
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