If a circle c, whose radius is 3, touches externally the circle x2 + y2 + 2x - 4y - 4 = 0 at the point (2, 2), then the length of the intercept cut by this circle c, on the x-axis, is equal to
Answers
Answer:
2√5
Step-by-step explanation:
Hi,
Let the center of the circle c be 'A
Given radius of the circle c as 3,
Also, given that the circle c touches x² + y² + 2x - 4y - 4 = 0 at point D(2, 2).
Center of the given circle is B (-1, 2) and the radius of the given circle is
√1² + 2² + 4 = 3.
Since 2 circles touch each other externally at D(2, 2) , center A , D and B lie
on a straight line.
But equation of line BD is y = 2 since both the points B and D have same y-
coordinate as 2.
Thus, center of circle c will also have y-coordinate 2.
Now, since circles touch each other externally, distance between center's
should be equal to sum of their radii,
hence A will have its coordinates as (5, 2).
Hence the equation of circle c will be
(x - 5)² + (y - 2)² = 3²
Now to find the intercept of circle on x- axis , let y = 0
⇒ (x - 5)² + 4 = 9
⇒ (x - 5)² = 5
⇒ x - 5 = ±√5
⇒ x = 5 ± √5
Let x₁ = 5 + √5 and x₂ = 5 - √5
Hence , the x-intercept will be x₁ - x₂ = 2√5
Hope, it helped !
Answer:2√5 is the answer