Question
The circle \( C \) has equation \( x^{2}+y^{2}-20 x-24 y+195=0 \) The centre of \( C \) is at the point \( M(10,12) \). \( N \) is the point with coordinates \( (25,32) \). Find the length of the line \( M N \).
Ask by Hardy Allan. in Singapore
Jan 24,2025
Real Tutor Solution
Tutor-Verified Answer
Answer
The length of line \( MN \) is \( 25 \) units.
Solution
To find the length of the line segment \( MN \) between points \( M(10, 12) \) and \( N(25, 32) \), we can use the distance formula:
\[
\text{Distance} = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2}
\]
Plugging in the coordinates:
\[
\text{Distance} = \sqrt{(25 - 10)^2 + (32 - 12)^2}
\]
\[
\text{Distance} = \sqrt{15^2 + 20^2}
\]
\[
\text{Distance} = \sqrt{225 + 400}
\]
\[
\text{Distance} = \sqrt{625}
\]
\[
\text{Distance} = 25
\]
**Answer:** The length of line \( MN \) is \( 25 \) units.
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Simplify this solution Bonus Knowledge
To find the length of the line segment \( MN \), we can use the distance formula, which is given by: \[ d = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2} \] In this case, the coordinates of point \( M \) are \( (10, 12) \) and the coordinates of point \( N \) are \( (25, 32) \). Now, we can substitute these coordinates into the distance formula: \[ d = \sqrt{(25 - 10)^2 + (32 - 12)^2} \] Calculating the differences: \[ d = \sqrt{(15)^2 + (20)^2} \] Calculating the squares: \[ d = \sqrt{225 + 400} \] Adding the two values: \[ d = \sqrt{625} \] Taking the square root: \[ d = 25 \] Thus, the length of the line segment \( MN \) is \( 25 \).
