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The chord of the hyperbola (X2/a2)-(Y2/b2)=1,whose equation is xcos alpha ysin alpha=P,subtended a right angle at it centre and it always touches a circle of radius.?
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The chord of the hyperbola (X2/a2)-(Y2/b2)=1,whose equation is xcos al...
Understanding the Hyperbola
The given hyperbola is defined by the equation (X²/a²) - (Y²/b²) = 1. This represents a hyperbola centered at the origin, opening horizontally.
Chord Equation
The chord of the hyperbola is represented by the equation x cos α + y sin α = P, where α is the angle subtended at the center.
Right Angle Subtended at the Center
For a chord to subtend a right angle at the center of the hyperbola, the following condition must be satisfied:
- The slopes of the lines forming the chord should multiply to -1.
This condition ensures that the angle between the two lines equals 90 degrees.
Circle Touching the Hyperbola
The chord always touches a circle of radius, which can be derived from the properties of the hyperbola and the angle α. The radius of the circle can be represented as:
- Radius = b / √(sin² α + (a²/b²) cos² α)
This shows that the radius of the circle depends on the parameters a, b, and the angle α.
Conclusion
In summary, the chord of the hyperbola (X²/a²) - (Y²/b²) = 1, given by the equation x cos α + y sin α = P, subtends a right angle at the center and always touches a circle of radius determined by the relationship of a, b, and the angle α. This relationship highlights the geometric properties connecting hyperbolas and circles.
The given hyperbola is defined by the equation (X²/a²) - (Y²/b²) = 1. This represents a hyperbola centered at the origin, opening horizontally.
Chord Equation
The chord of the hyperbola is represented by the equation x cos α + y sin α = P, where α is the angle subtended at the center.
Right Angle Subtended at the Center
For a chord to subtend a right angle at the center of the hyperbola, the following condition must be satisfied:
- The slopes of the lines forming the chord should multiply to -1.
This condition ensures that the angle between the two lines equals 90 degrees.
Circle Touching the Hyperbola
The chord always touches a circle of radius, which can be derived from the properties of the hyperbola and the angle α. The radius of the circle can be represented as:
- Radius = b / √(sin² α + (a²/b²) cos² α)
This shows that the radius of the circle depends on the parameters a, b, and the angle α.
Conclusion
In summary, the chord of the hyperbola (X²/a²) - (Y²/b²) = 1, given by the equation x cos α + y sin α = P, subtends a right angle at the center and always touches a circle of radius determined by the relationship of a, b, and the angle α. This relationship highlights the geometric properties connecting hyperbolas and circles.
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The chord of the hyperbola (X2/a2)-(Y2/b2)=1,whose equation is xcos al...
The chord of the hyperbola (X2/a2)-(Y2/b2)=1,whose equation is xcos alpha ysin alpha=P,subtended a right angle at it centre and it always touches a circle of radius.?
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The chord of the hyperbola (X2/a2)-(Y2/b2)=1,whose equation is xcos alpha ysin alpha=P,subtended a right angle at it centre and it always touches a circle of radius.? for JEE 2025 is part of JEE preparation. The Question and answers have been prepared according to the JEE exam syllabus. Information about The chord of the hyperbola (X2/a2)-(Y2/b2)=1,whose equation is xcos alpha ysin alpha=P,subtended a right angle at it centre and it always touches a circle of radius.? covers all topics & solutions for JEE 2025 Exam. Find important definitions, questions, meanings, examples, exercises and tests below for The chord of the hyperbola (X2/a2)-(Y2/b2)=1,whose equation is xcos alpha ysin alpha=P,subtended a right angle at it centre and it always touches a circle of radius.?.
The chord of the hyperbola (X2/a2)-(Y2/b2)=1,whose equation is xcos alpha ysin alpha=P,subtended a right angle at it centre and it always touches a circle of radius.? for JEE 2025 is part of JEE preparation. The Question and answers have been prepared according to the JEE exam syllabus. Information about The chord of the hyperbola (X2/a2)-(Y2/b2)=1,whose equation is xcos alpha ysin alpha=P,subtended a right angle at it centre and it always touches a circle of radius.? covers all topics & solutions for JEE 2025 Exam. Find important definitions, questions, meanings, examples, exercises and tests below for The chord of the hyperbola (X2/a2)-(Y2/b2)=1,whose equation is xcos alpha ysin alpha=P,subtended a right angle at it centre and it always touches a circle of radius.?.
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