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A and B are two distinct points, Locus of a point P satisfying |PA| + |PB| = 2k, a constant is
- a)nothing can be said
- b)the line segment [AB]
- c)an ellipse
- d)a hyperbola
Correct answer is option 'A'. Can you explain this answer?
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Most Upvoted Answer
A and B are two distinct points, Locus of a point P satisfying|PA|+|PB...
I think the locus of P is an ellipse because if yo
u take any point on the ellipse then the sum of distance from focii to that
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A and B are two distinct points, Locus of a point P satisfying|PA|+|PB...
Explanation:
To find the locus of a point P satisfying the condition |PA| |PB| = 2k, we need to analyze the given equation and determine the shape formed by all points that satisfy it.
Understanding the Equation:
The equation |PA| |PB| = 2k represents a relationship between the distances of point P from two fixed points A and B. The product of the distances |PA| and |PB| is constant, equal to 2k.
Case 1: When A and B are coincident points:
If points A and B coincide, the equation becomes |PA|^2 = 2k. This equation represents a circle with center A (or B) and radius √(2k).
Case 2: When A and B are distinct points:
In this case, we need to analyze the equation further to determine the shape formed by the locus of point P.
Exploring the Locus:
Let's consider a few scenarios to understand the locus.
- When P lies on the line segment AB, |PA| + |PB| = |AB|. Substituting this into the equation, we get |AB| = 2k. Thus, all points P on the line segment AB satisfy the equation.
- When P lies outside the line segment AB, |PA| + |PB| > |AB|. In this case, the equation |PA| |PB| = 2k cannot be satisfied.
- When P lies inside the line segment AB, |PA| + |PB| < |ab|.="" similarly,="" the="" equation="" |pa|="" |pb|="2k" cannot="" be="" />
Conclusion:
From the analysis, we can conclude that the locus of point P satisfying the equation |PA| |PB| = 2k, when A and B are distinct points, is an empty set. In other words, there are no points that satisfy the given equation when A and B are distinct.
Hence, the correct answer is option 'A' - nothing can be said.
To find the locus of a point P satisfying the condition |PA| |PB| = 2k, we need to analyze the given equation and determine the shape formed by all points that satisfy it.
Understanding the Equation:
The equation |PA| |PB| = 2k represents a relationship between the distances of point P from two fixed points A and B. The product of the distances |PA| and |PB| is constant, equal to 2k.
Case 1: When A and B are coincident points:
If points A and B coincide, the equation becomes |PA|^2 = 2k. This equation represents a circle with center A (or B) and radius √(2k).
Case 2: When A and B are distinct points:
In this case, we need to analyze the equation further to determine the shape formed by the locus of point P.
Exploring the Locus:
Let's consider a few scenarios to understand the locus.
- When P lies on the line segment AB, |PA| + |PB| = |AB|. Substituting this into the equation, we get |AB| = 2k. Thus, all points P on the line segment AB satisfy the equation.
- When P lies outside the line segment AB, |PA| + |PB| > |AB|. In this case, the equation |PA| |PB| = 2k cannot be satisfied.
- When P lies inside the line segment AB, |PA| + |PB| < |ab|.="" similarly,="" the="" equation="" |pa|="" |pb|="2k" cannot="" be="" />
Conclusion:
From the analysis, we can conclude that the locus of point P satisfying the equation |PA| |PB| = 2k, when A and B are distinct points, is an empty set. In other words, there are no points that satisfy the given equation when A and B are distinct.
Hence, the correct answer is option 'A' - nothing can be said.
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