The angle between the two tangents from the origin to the circle (x –7)2+ (y + 1)2= 25 equalsa)π/2b)π/3c)π/4d)NoneCorrect answer is option 'A'. Can you explain this answer?

Question

The angle between the two tangents from the origin to the circle (x &ndash;7)2+ (y + 1)2= 25 equalsa)&pi;/2b)&pi;/3c)&pi;/4d)NoneCorrect answer is option 'A'. Can you explain this answer?

Answer

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)|)

Answer

Let tangent from origin be y = mx
Using the condition of tangency, we get