Quantrex Quantrex AcademyJEE · NEET · NDA PYQs with solutions Open app
NTA Abhyas 2020
Exam
NTA Abhyas
Year
2020
Subject
Mathematics
Chapter
Hyperbola

NTA Abhyas 2020 · Mathematics · Hyperbola

The equation of a hyperbola is x^ 2 a^ 2 - y^ 2 b^ 2 = 1 . If P ( 2...

The equation of a hyperbola is x2a2−y2b2=1\dfrac{x^{2}}{a^{2}} - \dfrac{y^{2}}{b^{2}} = 1. If P(2,5)isapointP \left(\sqrt{2}, \sqrt{5}\right) is a point from which perpendicular tangents can be drawn to the hyperbola and distance between both the foci of the hyperbola is 10, then its eccentricity is

Options

  1. A32\dfrac{3}{2}
  2. B54\dfrac{5}{4}
  3. C169\dfrac{16}{9}
  4. D43\dfrac{4}{3}

Correct answer

B. 54\dfrac{5}{4}

Step-by-step solution

∵P∵ P lies on director circle x2+y2=a2−b2x^{2} + y^{2} = a^{2} - b^{2}
∴a2−b2=7∴a^{2} - b^{2} = 7 …(i)
Also, 2ae=10⇒ae=52ae = 10 ⇒ ae = 5 …(ii)
Now, e2=a2+b2a2⇒a2+b2=25e^{2} = \dfrac{a^{2} + b^{2}}{a^{2}} ⇒ a^{2} + b^{2} = 25 …(iii)
From (i) & (iii), we get,
a2=16,b2=9a^{2} = 16, b^{2} = 9
e=2516=54e = \sqrt{\dfrac{25}{16}} = \dfrac{5}{4}
Practice Hyperbola on Quantrex Academy →

More from Hyperbola

If the eccentricity e of the hyperbola (x^2)/(a^2) - (y^2)/(b^2) = 1 , passing through… JEE Main 2026Let the eccentricity e of a hyperbola satisfy the equation 6e^2 - 11e + 3 = 0 . If the… JEE Main 2026Let H: (x^2)/(a^2)-(y^2)/(b^2)=1 be a hyperbola such that the distance between its foci… JEE Main 2026Let O be the origin, and P and Q be two points on the rectangular hyperbola xy = 12 such… JEE Main 2026For some θ ∈(0, (π)/(2)) , let the eccentricity and the length of the latus rectum of the… JEE Main 2026Let PQ be a chord of the hyperbola x^ 2 4 - y^ 2 b^ 2 =1 , perpendicular to the x -axis… JEE Main 2026Let the domain of the function f(x)= _ 3 _ 5 _ 7 (9 x-x^ 2 -13) be the interval (m, n) .… JEE Main 2026Let P(10,2 15 ) be a point on the hyperbola x^ 2 a^ 2 - y^ 2 ~b^ 2 =1 , whose foci are S… JEE Main 2026If the line α x+2 y=1 , where α ∈ ℝ , does not meet the hyperbola x^ 2 -9 y^ 2 =9 , then… JEE Main 2026Let the foci of a hyperbola coincide with the foci of the ellipse x^ 2 36 + y^ 2 16 =1 .… JEE Main 2026Let e_1 and e_2 be the eccentricities of the ellipse x^2 ~b^2 + y^2 25 =1 and the… JEE Main 2025Let the lengths of the transverse and conjugate axes of a hyperbola in standard form be… JEE Main 2025Consider the hyperbola (x^2)/(a^2)-(y^2)/(b^2)=1 having one of its focus at P(-3,0) . If… JEE Main 2025Let the sum of the focal distances of the point P(4,3) on the hyperbola H: x^2 a^2 - y^2… JEE Main 2025If the equation of the hyperbola with foci (4,2) and (8,2) is 3 x^2-y^2-α x+β y+γ=0… JEE Main 2025Let the product of the focal distances of the point P(4,2 3 ) on the hyperbola H: x^2 a^2… JEE Main 2025 Full Hyperbola list Hyperbola JEE Maths PYQsJEE Mathematics PYQs All NTA Abhyas JEE Main PYQs