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The line y=mx+c will be a normal to the ...

The line `y=mx+c` will be a normal to the circle with radius `r` and centre at `(a,b)` if

A

`a=mb+c`

B

`b=ma+c`

C

`c=ma+b`

D

`c=mb+a`

Text Solution

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The correct Answer is:
To determine the condition under which the line \( y = mx + c \) is a normal to the circle with radius \( r \) and center at \( (a, b) \), we can follow these steps: ### Step 1: Understand the relationship between the line and the circle A line is said to be normal to a circle if it is perpendicular to the radius at the point of tangency. This means that the line must pass through the center of the circle. ### Step 2: Substitute the center of the circle into the line equation The center of the circle is given as \( (a, b) \). If the line \( y = mx + c \) is normal to the circle, it must pass through this center. Therefore, we can substitute \( x = a \) and \( y = b \) into the line equation: \[ b = ma + c \] ### Step 3: Rearrange the equation From the equation \( b = ma + c \), we can rearrange it to express \( c \): \[ c = b - ma \] ### Conclusion Thus, the line \( y = mx + c \) will be a normal to the circle with radius \( r \) and center at \( (a, b) \) if: \[ b = ma + c \] This indicates that the line must pass through the center of the circle.
To determine the condition under which the line \( y = mx + c \) is a normal to the circle with radius \( r \) and center at \( (a, b) \), we can follow these steps: ### Step 1: Understand the relationship between the line and the circle A line is said to be normal to a circle if it is perpendicular to the radius at the point of tangency. This means that the line must pass through the center of the circle. ### Step 2: Substitute the center of the circle into the line equation The center of the circle is given as \( (a, b) \). If the line \( y = mx + c \) is normal to the circle, it must pass through this center. Therefore, we can substitute \( x = a \) and \( y = b \) into the line equation: ...
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