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Let a and b be positive real numbers such that a > 1 and b < a. Let P be a point in the first quadrant that lies on the hyperbola  Suppose the tangent to the hyperbola at P passes through the point (1,0), and suppose the normal to the hyperbola at P cuts off equal intercepts on the coordinate axes. Let Δ denote the area of the triangle formed by the tangent at P, the normal at P and the  x-axis. If e denotes the eccentricity of the hyperbola, then which of the following statements is/are TRUE?
  • a)
    1 < e < 2 
  • b)
    √2 < e < 2  
  • c)
    Δ = a4
  • d)
    Δ = b4
Correct answer is option 'A,D'. Can you explain this answer?
Verified Answer
Let aand bbe positive real numbers such that a> 1 and b< a. Let ...

Since Normal at point P makes equal intercept on co-ordinate axes, therefore slope of Normal = –1
Hence slope of tangent = 1  
Equation of tangent
y – 0 = 1(x–1)  
y = x – 1
Equation of tangent at (x1y1

x – y = 1 (equation of Tangent)  on comparing x1 = a2, y1 – b2
Also a2 – b2 = 1  ...(1)  
Now equation of normal at (x1y1) = (a21b2)
y – b2 = – 1(x – a2)  
x + y = a2 + b2  ...(Normal)  
point of intersection with x-axis is (a2 + b2

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Let aand bbe positive real numbers such that a> 1 and b< a. Let Pbe a point in the first quadrantthat lies on the hyperbolaSuppose the tangent to the hyperbola at Ppasses through the point (1,0), and suppose the normal to the hyperbola at Pcuts off equal intercepts on the coordinate axes. Let Δ denote the area of the triangle formed by the tangent at P, the normal at Pand the x-axis. If edenotes the eccentricity of the hyperbola, then which of the following statements is/areTRUE?a)1 < e< 2b)√2 < e< 2c)Δ = a4d)Δ = b4Correct answer is option 'A,D'. Can you explain this answer?
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Let aand bbe positive real numbers such that a> 1 and b< a. Let Pbe a point in the first quadrantthat lies on the hyperbolaSuppose the tangent to the hyperbola at Ppasses through the point (1,0), and suppose the normal to the hyperbola at Pcuts off equal intercepts on the coordinate axes. Let Δ denote the area of the triangle formed by the tangent at P, the normal at Pand the x-axis. If edenotes the eccentricity of the hyperbola, then which of the following statements is/areTRUE?a)1 < e< 2b)√2 < e< 2c)Δ = a4d)Δ = b4Correct answer is option 'A,D'. Can you explain this answer? for JEE 2024 is part of JEE preparation. The Question and answers have been prepared according to the JEE exam syllabus. Information about Let aand bbe positive real numbers such that a> 1 and b< a. Let Pbe a point in the first quadrantthat lies on the hyperbolaSuppose the tangent to the hyperbola at Ppasses through the point (1,0), and suppose the normal to the hyperbola at Pcuts off equal intercepts on the coordinate axes. Let Δ denote the area of the triangle formed by the tangent at P, the normal at Pand the x-axis. If edenotes the eccentricity of the hyperbola, then which of the following statements is/areTRUE?a)1 < e< 2b)√2 < e< 2c)Δ = a4d)Δ = b4Correct answer is option 'A,D'. Can you explain this answer? covers all topics & solutions for JEE 2024 Exam. Find important definitions, questions, meanings, examples, exercises and tests below for Let aand bbe positive real numbers such that a> 1 and b< a. Let Pbe a point in the first quadrantthat lies on the hyperbolaSuppose the tangent to the hyperbola at Ppasses through the point (1,0), and suppose the normal to the hyperbola at Pcuts off equal intercepts on the coordinate axes. Let Δ denote the area of the triangle formed by the tangent at P, the normal at Pand the x-axis. If edenotes the eccentricity of the hyperbola, then which of the following statements is/areTRUE?a)1 < e< 2b)√2 < e< 2c)Δ = a4d)Δ = b4Correct answer is option 'A,D'. Can you explain this answer?.
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