Quantrex Quantrex AcademyJEE · NEET · NDA PYQs with solutions Open app
Highly selective Backlog Qs for JEE MainMathematicsHyperbola

Let P ( 4 ), Q ( 5 4 ), R ( 3 4 ), T ( 7 4 ) be the points on the hyperbola x^2-4 y^2-4=0 in the parametric form. Then the area of the quadrilateral P Q R T is (in square units)

Options

  1. A4 2
  2. B16 2
  3. C32 2
  4. D8 2

Correct answer

D. 8 2

Step-by-step solution

Equation of hyperbola x^2-4 y^2-4=0 x^2 4 - y^2 1 =1 Then, a=2, b=1 Parametric point on hyperbola =(a , b ) Then, P ( 4 )= (2 ( 4 ), 1 ( 4 ) ) =(2 2 , 1) Similarly, Q ( 5 4 )= (2 ( 5 4 ), 1 ( 5 4 ) ) aligned & =(-2 2 , 1) & R ( 3 4 )=(-2 2 ,-1) & T ( 7 4 )=(2 2 ,-1) aligned Area of quadrilateral PQRT(sq. unit) aligned & = 1 2 (x₁-x₃ ) (y₂-y₄ )- (x₂-x₄ ) (y₁-y₃ ) & = 1 2 (2 2 +2 2 )(1+1)-(-2 2 -2 2 )(1+1) & = 1 2 4 2 2+4 2 2 & = 1 2 16 2 =8 2 aligned So, option (d) is correct.

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 (6, 4 3 ) , satisfies 15(e^2 + 1) = 34e , then the length of the latus rectum of the hyperbola x^2 b^ 2026Let the eccentricity e of a hyperbola satisfy the equation 6e^2 - 11e + 3 = 0 . If the foci of the hyperbola are (3, 5) and (3, -4) , then the length of its latus rectum is : 2026Let H: x^2 a^2 - y^2 b^2 =1 be a hyperbola such that the distance between its foci is 6 and the distance between its directrices is 8 3 . If the line x= intersects the hyperbola H 2026Let O be the origin, and P and Q be two points on the rectangular hyperbola xy = 12 such that the mid point of the line segment PQ is ( 1 2 , - 1 2 ) . Then the area of the triangl 2026For some (0, 2 ) , let the eccentricity and the length of the latus rectum of the hyperbola x²-y² ² =8 be e₁ and l₁ , respectively, and let the eccentricity and the length of the l 2026Let PQ be a chord of the hyperbola x² 4 - y² b² =1 , perpendicular to the x -axis such that OPQ is an equilateral triangle, O being the centre of the hyperbola. If the eccentricity 2026Let the domain of the function f(x)= ₃ ₅ ₇ (9 x-x²-13 ) be the interval ( m , n ) . Let the hyperbola x² a ² - y² ~b ² =1 have eccentricity n 3 and the length of the latus rectum 8 2026Let P (10,2 15 ) be a point on the hyperbola x² a ² - y² ~b ² =1 , whose foci are S and S ^ . If the length of its latus rectum is 8, then the square of the area of PSS ^ is equal 2026 Full Hyperbola list All Highly selective Backlog Qs for JEE Main PYQs