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The angle between the two tangents from the origin to the circle (x – 7)2 + (y + 1)2 = 25 equals
- a)π/2
- b)π/3
- c)π/4
- d)None
Correct answer is option 'A'. Can you explain this answer?
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Verified Answer
The angle between the two tangents from the origin to the circle (x &n...
Let tangent from origin be y = mx
Using the condition of tangency, we get
The angle between tangents = π/2
Using the condition of tangency, we get
The angle between tangents = π/2
Most Upvoted Answer
The angle between the two tangents from the origin to the circle (x &n...
To find the angle between the two tangents from the origin to the circle, we can use the fact that the angle between a tangent and a radius at the point of tangency is always 90 degrees.
Let's assume the coordinates of the center of the circle are (h, k) and the radius is r.
The equation of the circle is given by:
(x - h)^2 + (y - k)^2 = r^2
Since the origin is (0, 0), the equation of the line passing through the origin and the center of the circle can be written as:
(y - k) = m(x - h)
Substituting x = 0 and y = 0, we get:
(-k) = mh
Simplifying, we get:
m = -k/h
The slope of a line perpendicular to m is the negative reciprocal of m. So, the slope of the tangent lines is:
m_perpendicular = -1/m = -1/(-k/h) = h/k
The angle between the two tangents can be found using the formula for the angle between two lines with slopes m1 and m2:
tan(theta) = |(m1 - m2) / (1 + m1 * m2)|
Substituting m1 = h/k and m2 = -h/k, we get:
tan(theta) = |(h/k + h/k) / (1 + h/k * -h/k)| = |(2h/k) / (1 - (h^2/k^2))|
Simplifying, we get:
tan(theta) = |(2h/k) / ((k^2 - h^2)/k^2)| = |2h / (k^2 - h^2)|
Therefore, the angle between the two tangents from the origin to the circle is given by:
theta = atan(|2h / (k^2 - h^2)|)
Let's assume the coordinates of the center of the circle are (h, k) and the radius is r.
The equation of the circle is given by:
(x - h)^2 + (y - k)^2 = r^2
Since the origin is (0, 0), the equation of the line passing through the origin and the center of the circle can be written as:
(y - k) = m(x - h)
Substituting x = 0 and y = 0, we get:
(-k) = mh
Simplifying, we get:
m = -k/h
The slope of a line perpendicular to m is the negative reciprocal of m. So, the slope of the tangent lines is:
m_perpendicular = -1/m = -1/(-k/h) = h/k
The angle between the two tangents can be found using the formula for the angle between two lines with slopes m1 and m2:
tan(theta) = |(m1 - m2) / (1 + m1 * m2)|
Substituting m1 = h/k and m2 = -h/k, we get:
tan(theta) = |(h/k + h/k) / (1 + h/k * -h/k)| = |(2h/k) / (1 - (h^2/k^2))|
Simplifying, we get:
tan(theta) = |(2h/k) / ((k^2 - h^2)/k^2)| = |2h / (k^2 - h^2)|
Therefore, the angle between the two tangents from the origin to the circle is given by:
theta = atan(|2h / (k^2 - h^2)|)
Community Answer
The angle between the two tangents from the origin to the circle (x &n...
Let tangent from origin be y = mx
Using the condition of tangency, we get
The angle between tangents = π/2
Using the condition of tangency, we get
The angle between tangents = π/2
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