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If a circle of radius b units with centre at (0, b) touches the line y = x — a√2 , then what is the value of b?
- a)2 + √2
- b)2-√2
- c)2√2
- d)√2
Correct answer is option 'A'. Can you explain this answer?
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If a circle of radius b units with centre at (0, b) touches the line y...
Distance from the centre to the point of line which touches circle is OM = radius
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If a circle of radius b units with centre at (0, b) touches the line y...
If a circle with radius b units and center at (0, b) touches the line y = x, then the distance between the center of the circle and the line y = x should be equal to the radius of the circle.
The equation of the line y = x can be written in the general form as x - y = 0.
The distance between a point (x₁, y₁) and a line Ax + By + C = 0 is given by the formula:
d = |Ax₁ + By₁ + C| / √(A² + B²)
In our case, the equation of the line y = x can be written as x - y = 0, so A = 1, B = -1, and C = 0.
The center of the circle is (0, b), so x₁ = 0 and y₁ = b.
Substituting these values into the distance formula, we get:
d = |1(0) + (-1)(b) + 0| / √(1² + (-1)²)
= |-b| / √(1 + 1)
= b / √2
Since the distance between the center of the circle and the line y = x should be equal to the radius of the circle (b), we have:
b / √2 = b
Multiplying both sides of the equation by √2, we get:
b = b√2
Dividing both sides of the equation by b, we get:
1 = √2
However, 1 is not equal to √2. Therefore, the circle with radius b units and center at (0, b) cannot touch the line y = x.
The equation of the line y = x can be written in the general form as x - y = 0.
The distance between a point (x₁, y₁) and a line Ax + By + C = 0 is given by the formula:
d = |Ax₁ + By₁ + C| / √(A² + B²)
In our case, the equation of the line y = x can be written as x - y = 0, so A = 1, B = -1, and C = 0.
The center of the circle is (0, b), so x₁ = 0 and y₁ = b.
Substituting these values into the distance formula, we get:
d = |1(0) + (-1)(b) + 0| / √(1² + (-1)²)
= |-b| / √(1 + 1)
= b / √2
Since the distance between the center of the circle and the line y = x should be equal to the radius of the circle (b), we have:
b / √2 = b
Multiplying both sides of the equation by √2, we get:
b = b√2
Dividing both sides of the equation by b, we get:
1 = √2
However, 1 is not equal to √2. Therefore, the circle with radius b units and center at (0, b) cannot touch the line y = x.
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