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The locus of a point P(α, β) moving under the condition that the line y = αx + β is a tangent to the hyperbola (x2/a) - (y2/b2) = 1 is
- a)a circle
- b)an ellipse
- c)a hyperbola
- d)a parabola
Correct answer is option 'C'. Can you explain this answer?
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The locus of a point P(α, β) moving under the condition tha...
1. **Understanding the problem**
The locus of a point P(α, β) moving under the condition that the line y = αx + β is a tangent to the hyperbola (x2/a) - (y2/b2) = 1 needs to be determined.
2. **Deriving the equation**
When the line y = αx + β is a tangent to the hyperbola, the point of tangency satisfies both the equation of the hyperbola and the equation of the line. By solving these two equations simultaneously, we can find the point of tangency.
3. **Finding the locus**
After obtaining the point of tangency, we can express it in terms of α and β. The locus of this point P(α, β) will give us the required equation.
4. **Analyzing the locus**
Upon analyzing the equation obtained for the locus, it can be seen that it represents a hyperbola. This is because the locus of a point moving under the given condition forms a hyperbolic curve.
5. **Conclusion**
Therefore, the correct answer is option 'C', a hyperbola. The locus of the point moving under the condition specified forms a hyperbola, as determined through the derivation and analysis of the problem.
The locus of a point P(α, β) moving under the condition that the line y = αx + β is a tangent to the hyperbola (x2/a) - (y2/b2) = 1 needs to be determined.
2. **Deriving the equation**
When the line y = αx + β is a tangent to the hyperbola, the point of tangency satisfies both the equation of the hyperbola and the equation of the line. By solving these two equations simultaneously, we can find the point of tangency.
3. **Finding the locus**
After obtaining the point of tangency, we can express it in terms of α and β. The locus of this point P(α, β) will give us the required equation.
4. **Analyzing the locus**
Upon analyzing the equation obtained for the locus, it can be seen that it represents a hyperbola. This is because the locus of a point moving under the given condition forms a hyperbolic curve.
5. **Conclusion**
Therefore, the correct answer is option 'C', a hyperbola. The locus of the point moving under the condition specified forms a hyperbola, as determined through the derivation and analysis of the problem.
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The locus of a point P(α, β) moving under the condition that the line y = αx + β is a tangent to the hyperbola (x2/a) - (y2/b2) = 1 isa)a circleb)an ellipsec)a hyperbolad)a parabolaCorrect answer is option 'C'. Can you explain this answer?
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