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The number of common tangents to the circles x2 + y2 = 4 and x2 + y2 – 6x – 8y = 24 is (1998 - 2 Marks)
- a)0
- b)1
- c)3
- d)4
Correct answer is option 'B'. Can you explain this answer?
Verified Answer
The number of common tangents to the circles x2 + y2 = 4 and x2 + y2 &...
x2 + y2 = 4 (given) Centre C1 ≡ (0, 0) and R1 = 2.
Also for circle x2 + y2 – 6x – 8y – 24 = 0 C2 ≡ (3, 4) and R2 = 7.
Again C1 C2 = 5 = R2 – R1
Also for circle x2 + y2 – 6x – 8y – 24 = 0 C2 ≡ (3, 4) and R2 = 7.
Again C1 C2 = 5 = R2 – R1
Therefore, the given circles touch internally such that they can have just one common tangent at the point of contact.
Most Upvoted Answer
The number of common tangents to the circles x2 + y2 = 4 and x2 + y2 &...
Common Tangents Between Circles
The number of common tangents between two circles can be determined by analyzing their relative positions. In this case, we have two circles with equations x2 + y2 = 4 and x2 + y2 – 6x – 8y = 24.
Circle 1: x2 + y2 = 4
This circle has a radius of √4 = 2 and is centered at the origin (0,0).
Circle 2: x2 + y2 – 6x – 8y = 24
This circle can be rewritten as (x - 3)2 + (y - 4)2 = 49, which means it has a radius of 7 and is centered at (3,4).
Analysis
Since the circles do not intersect or enclose each other, they are externally tangent. In this case, they will have one common tangent.
Number of Common Tangents
So, the answer is option B) 1 common tangent between the two circles.
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