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If the mid-point of the line segment joining the points A (3, 4) and B (a, 4) is P (x, y) and x + y - 20 = 0, then find the value of a?
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If the mid-point of the line segment joining the points A (3, 4) and B...
Finding the Midpoint P
To find the midpoint P of the line segment joining points A(3, 4) and B(a, 4), we use the midpoint formula:
- Midpoint Formula:
\[ P(x, y) = \left( \frac{x_1 + x_2}{2}, \frac{y_1 + y_2}{2} \right) \]
where \( (x_1, y_1) \) and \( (x_2, y_2) \) are the coordinates of points A and B.
- For points A(3, 4) and B(a, 4):
- \( x_1 = 3, y_1 = 4 \)
- \( x_2 = a, y_2 = 4 \)
- Applying the formula:
\[ P(x, y) = \left( \frac{3 + a}{2}, \frac{4 + 4}{2} \right) \]
\[ P(x, y) = \left( \frac{3 + a}{2}, 4 \right) \]
Thus, the coordinates of midpoint P are:
\( P\left(\frac{3 + a}{2}, 4\right) \)
Using the Given Equation
We know from the problem statement that:
\( x + y - 20 = 0 \)
Substituting the coordinates of P into the equation:
- Let \( x = \frac{3 + a}{2} \) and \( y = 4 \).
- Substituting these values yields:
\[ \frac{3 + a}{2} + 4 - 20 = 0 \]
- Simplifying the equation:
\[ \frac{3 + a}{2} - 16 = 0 \]
\[ \frac{3 + a}{2} = 16 \]
\[ 3 + a = 32 \]
\[ a = 32 - 3 \]
\[ a = 29 \]
Conclusion
The value of a is:
\( a = 29 \).
To find the midpoint P of the line segment joining points A(3, 4) and B(a, 4), we use the midpoint formula:
- Midpoint Formula:
\[ P(x, y) = \left( \frac{x_1 + x_2}{2}, \frac{y_1 + y_2}{2} \right) \]
where \( (x_1, y_1) \) and \( (x_2, y_2) \) are the coordinates of points A and B.
- For points A(3, 4) and B(a, 4):
- \( x_1 = 3, y_1 = 4 \)
- \( x_2 = a, y_2 = 4 \)
- Applying the formula:
\[ P(x, y) = \left( \frac{3 + a}{2}, \frac{4 + 4}{2} \right) \]
\[ P(x, y) = \left( \frac{3 + a}{2}, 4 \right) \]
Thus, the coordinates of midpoint P are:
\( P\left(\frac{3 + a}{2}, 4\right) \)
Using the Given Equation
We know from the problem statement that:
\( x + y - 20 = 0 \)
Substituting the coordinates of P into the equation:
- Let \( x = \frac{3 + a}{2} \) and \( y = 4 \).
- Substituting these values yields:
\[ \frac{3 + a}{2} + 4 - 20 = 0 \]
- Simplifying the equation:
\[ \frac{3 + a}{2} - 16 = 0 \]
\[ \frac{3 + a}{2} = 16 \]
\[ 3 + a = 32 \]
\[ a = 32 - 3 \]
\[ a = 29 \]
Conclusion
The value of a is:
\( a = 29 \).
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If the mid-point of the line segment joining the points A (3, 4) and B (a, 4) is P (x, y) and x + y - 20 = 0, then find the value of a?
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If the mid-point of the line segment joining the points A (3, 4) and B (a, 4) is P (x, y) and x + y - 20 = 0, then find the value of a? for UPSC 2024 is part of UPSC preparation. The Question and answers have been prepared according to the UPSC exam syllabus. Information about If the mid-point of the line segment joining the points A (3, 4) and B (a, 4) is P (x, y) and x + y - 20 = 0, then find the value of a? covers all topics & solutions for UPSC 2024 Exam. Find important definitions, questions, meanings, examples, exercises and tests below for If the mid-point of the line segment joining the points A (3, 4) and B (a, 4) is P (x, y) and x + y - 20 = 0, then find the value of a?.
If the mid-point of the line segment joining the points A (3, 4) and B (a, 4) is P (x, y) and x + y - 20 = 0, then find the value of a? for UPSC 2024 is part of UPSC preparation. The Question and answers have been prepared according to the UPSC exam syllabus. Information about If the mid-point of the line segment joining the points A (3, 4) and B (a, 4) is P (x, y) and x + y - 20 = 0, then find the value of a? covers all topics & solutions for UPSC 2024 Exam. Find important definitions, questions, meanings, examples, exercises and tests below for If the mid-point of the line segment joining the points A (3, 4) and B (a, 4) is P (x, y) and x + y - 20 = 0, then find the value of a?.
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