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Two circles of equal radii touch externally at a point P. From a point T on the tangent at P, tangents TQ and TR are drawn to the circles with points of contact Q and R respectively. The relation of TQ and TR is (SSC CGL 1st Sit. 2013)
- a)TQ < TR
- b)TQ > TR
- c)TQ = 2TR
- d)TQ = TR
Correct answer is option 'D'. Can you explain this answer?
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Verified Answer
Two circles of equal radii touch externally at a point P. From a point...
Let two circles with centre O and O' touches each other at point P. From point T tangent TQ and TR are drawn on two circles equal radius.
As we know two tangent drawn from a external points are always equal
So, TQ = TP ...(i)
and TP = TR ...(ii)
from (i) and (ii), TQ = TP = TR
Most Upvoted Answer
Two circles of equal radii touch externally at a point P. From a point...
Let two circles with centre O and O' touches each other at point P. From point T tangent TQ and TR are drawn on two circles equal radius.
As we know two tangent drawn from a external points are always equal
So, TQ = TP ...(i)
and TP = TR ...(ii)
from (i) and (ii), TQ = TP = TR
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Community Answer
Two circles of equal radii touch externally at a point P. From a point...
Understanding the Geometry of the Problem
In this problem, we have two circles of equal radius that touch externally at point P. We also have a point T on the tangent at point P, from which we draw tangents TQ and TR to the two circles.
Key Properties of Tangents
- Tangents from an External Point: The length of the tangents drawn from an external point to a circle is equal.
- Equal Circles: Since both circles have equal radii and touch each other externally, the distance from point T to the center of each circle is the same.
Analyzing the Tangent Lengths
- Let O1 and O2: Denote the centers of the two circles as O1 and O2.
- Circle Radii: Let the radius of each circle be r.
- Distance from T: The distance from point T to the centers O1 and O2 will be equal due to the symmetry of the arrangement.
Equal Lengths of Tangents
- The length of the tangents from point T to the circles can be represented as:
- TQ (to circle O1)
- TR (to circle O2)
- Since T is equidistant from both centers (O1 and O2), the lengths TQ and TR will also be equal.
Conclusion
Thus, the correct relationship is:
- TQ = TR
This equality confirms that the lengths of the tangents drawn from an external point to two externally touching circles of equal radius are indeed equal, making option 'D' the correct choice.
In this problem, we have two circles of equal radius that touch externally at point P. We also have a point T on the tangent at point P, from which we draw tangents TQ and TR to the two circles.
Key Properties of Tangents
- Tangents from an External Point: The length of the tangents drawn from an external point to a circle is equal.
- Equal Circles: Since both circles have equal radii and touch each other externally, the distance from point T to the center of each circle is the same.
Analyzing the Tangent Lengths
- Let O1 and O2: Denote the centers of the two circles as O1 and O2.
- Circle Radii: Let the radius of each circle be r.
- Distance from T: The distance from point T to the centers O1 and O2 will be equal due to the symmetry of the arrangement.
Equal Lengths of Tangents
- The length of the tangents from point T to the circles can be represented as:
- TQ (to circle O1)
- TR (to circle O2)
- Since T is equidistant from both centers (O1 and O2), the lengths TQ and TR will also be equal.
Conclusion
Thus, the correct relationship is:
- TQ = TR
This equality confirms that the lengths of the tangents drawn from an external point to two externally touching circles of equal radius are indeed equal, making option 'D' the correct choice.
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Two circles of equal radii touch externally at a point P. From a point T on the tangent at P, tangents TQ and TR are drawn to the circles with points of contact Q and R respectively. The relation of TQ and TR is (SSC CGL 1st Sit. 2013)a)TQ < TRb)TQ > TRc)TQ = 2TRd)TQ = TRCorrect answer is option 'D'. Can you explain this answer?
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