DEEP STUDY OF COORDINATE GEOMETRY

Here’s a deeper dive into Coordinate Geometry with additional examples, real-life applications, and practical problem-solving techniques.

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Advanced Examples and Applications

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1. Using the Distance Formula

Problem: A triangle has vertices A(3,4)A(3, 4), B(7,1)B(7, 1), and C(5,6)C(5, 6). Determine whether the triangle is isosceles, scalene, or equilateral.

Solution:
Calculate the distances between each pair of vertices using the distance formula:

d=(x2−x1)2+(y2−y1)2.d = \sqrt{(x_2 – x_1)^2 + (y_2 – y_1)^2}.

• ABAB:

AB=(7−3)2+(1−4)2=42+(−3)2=16+9=25=5.AB = \sqrt{(7 – 3)^2 + (1 – 4)^2} = \sqrt{4^2 + (-3)^2} = \sqrt{16 + 9} = \sqrt{25} = 5.

• BCBC:

BC=(5−7)2+(6−1)2=(−2)2+52=4+25=29.BC = \sqrt{(5 – 7)^2 + (6 – 1)^2} = \sqrt{(-2)^2 + 5^2} = \sqrt{4 + 25} = \sqrt{29}.

• CACA:

CA=(5−3)2+(6−4)2=22+22=4+4=8.CA = \sqrt{(5 – 3)^2 + (6 – 4)^2} = \sqrt{2^2 + 2^2} = \sqrt{4 + 4} = \sqrt{8}.

Since all three sides are of different lengths (5,29,85, \sqrt{29}, \sqrt{8}), the triangle is scalene.

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2. Using the Section Formula

Problem: Find the coordinates of a point PP that divides the line segment joining A(2,3)A(2, 3) and B(8,7)B(8, 7) in the ratio 3:23:2.

Solution:
The section formula is:

P(x,y)=(mx2+nx1m+n,my2+ny1m+n).P(x, y) = \left(\frac{m x_2 + n x_1}{m+n}, \frac{m y_2 + n y_1}{m+n}\right).

Here, m=3m = 3, n=2n = 2, x1=2x_1 = 2, y1=3y_1 = 3, x2=8x_2 = 8, y2=7y_2 = 7.

x=3(8)+2(2)3+2=24+45=285=5.6.x = \frac{3(8) + 2(2)}{3 + 2} = \frac{24 + 4}{5} = \frac{28}{5} = 5.6. y=3(7)+2(3)3+2=21+65=275=5.4.y = \frac{3(7) + 2(3)}{3 + 2} = \frac{21 + 6}{5} = \frac{27}{5} = 5.4.

Answer: P(5.6,5.4)P(5.6, 5.4).

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3. Application in Navigation

Problem: A ship is at A(−3,4)A(-3, 4), and a lighthouse is at B(5,−2)B(5, -2). Find the shortest distance between the ship and the lighthouse.

Solution:
The shortest distance is the straight-line distance between AA and BB. Use the distance formula:

AB=(x2−x1)2+(y2−y1)2.AB = \sqrt{(x_2 – x_1)^2 + (y_2 – y_1)^2}.

Substitute x1=−3x_1 = -3, y1=4y_1 = 4, x2=5x_2 = 5, y2=−2y_2 = -2:

AB=(5−(−3))2+(−2−4)2=(5+3)2+(−6)2.AB = \sqrt{(5 – (-3))^2 + (-2 – 4)^2} = \sqrt{(5 + 3)^2 + (-6)^2}. AB=82+62=64+36=100=10.AB = \sqrt{8^2 + 6^2} = \sqrt{64 + 36} = \sqrt{100} = 10.

Answer: The shortest distance is 10 units10 \, \text{units}.

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4. Using the Area of a Triangle

Problem: Find the area of a triangle formed by the points A(1,1)A(1, 1), B(4,5)B(4, 5), and C(6,2)C(6, 2).

Solution:
Use the area formula:

Area=12∣x1(y2−y3)+x2(y3−y1)+x3(y1−y2)∣.\text{Area} = \frac{1}{2} \left| x_1(y_2 – y_3) + x_2(y_3 – y_1) + x_3(y_1 – y_2) \right|.

Substitute A(1,1)A(1, 1), B(4,5)B(4, 5), C(6,2)C(6, 2):

Area=12∣1(5−2)+4(2−1)+6(1−5)∣.\text{Area} = \frac{1}{2} \left| 1(5 – 2) + 4(2 – 1) + 6(1 – 5) \right|. =12∣1(3)+4(1)+6(−4)∣.= \frac{1}{2} \left| 1(3) + 4(1) + 6(-4) \right|. =12∣3+4−24∣=12∣−17∣=172.= \frac{1}{2} \left| 3 + 4 – 24 \right| = \frac{1}{2} \left| -17 \right| = \frac{17}{2}.

Answer: Area = 8.5 square units8.5 \, \text{square units}.

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Real-Life Applications of Coordinate Geometry

1. Navigation Systems (GPS):
   Coordinate geometry is used in GPS technology to find distances and routes between locations.
2. Urban Planning:
   Helps design layouts of cities, determining distances between landmarks or streets.
3. Astronomy:
   Used to locate celestial bodies in space.
4. Sports Analytics:
   Coordinates are used to track player movements and analyze strategies.
5. Computer Graphics:
   Shapes, animations, and 3D models are built using coordinate systems.
6. Robotics:
   Robots use coordinate geometry to navigate through their environments.

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Common Errors and How to Avoid Them

1. Sign Errors:
   • Always check the signs of coordinates before applying formulas, especially when working in different quadrants.
2. Formula Misuse:
   • Clearly understand whether to use the distance, section, or midpoint formula based on the problem.
3. Misrepresentation:
   • Plot points carefully on the graph to avoid errors in visualizing their positions.

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These advanced problems and applications showcase the importance of coordinate geometry in real-world scenarios and academic problem-solving. Let me know if you’d like to work on any other specific example or concept!

plz comment regarding your specific problem….