Quantrex Quantrex AcademyJEE · NEET · NDA PYQs with solutions Open app
JEE Advanced2025MathematicsEllipseActual

Let P (x₁, y₁ ) and Q (x₂, y₂ ) be two distinct points on the ellipse x^2 9 + y^2 4 =1 such that y₁>0 , and y₂>0 . Let C denote the circle x^2+y^2=9 , and M be the point (3,0) . Suppose the line x=x₁ intersects C at R , and the line x=x₂ intersects C at S , such that the y -coordinates of R and S are positive. Let R O M= 6 and S O M= 3 , where O denotes the origin (0,0) . Let |X Y| denote the length of the line segme

Options

  1. AThe equation of the line joining P and Q is 2 x+3 y=3(1+ 3 )
  2. BThe equation of the line joining P and Q is 2 x+y=3(1+ 3 )
  3. CIf N₂= (x₂, 0 ) , then 3 |N₂ Q |=2 |N₂ S |
  4. DIf N₁= (x₁, 0 ) , then 9 |N₁ P |=4 |N₁ R |

Correct answer

A. The equation of the line joining P and Q is 2 x+3 y=3(1+ 3 )

Step-by-step solution

aligned & P (3 30^ , 2 30^ ) ( 3 3 2 , 1 ) & Q (3 60^ , 2 60^ ) ( 3 2 , 3 ) & R ( 3 3 2 , 3 2 ), S ( 3 2 , 3 3 2 ) aligned Slope of PQ = m _ PQ = 3 -1 3 2 - 3 3 2 =- 2 3 Equation of line PQ y - 3 =- 2 3 ( x - 3 2 ) 2 x+3 y=3( 3 +1) option (A) is correct Now aligned & if N ₂= ( x ₂, 0 )= ( 3 2 , 0 ) & | N ₂ Q |= 3 and | N ₂ ~S |= 3 3 2 aligned 3 | ~N ₂ Q |=2 | ~N ₂ ~S | option (C) is correct Now, if N ₁= ( x ₁, 0 ) N ₁= ( 3 3 2 , 0 ) | N ₁ P |=1, | ~N ₁ R |= 3 2 option (D) is incorrect

Practice Ellipse on Quantrex Academy →

More from Ellipse

Consider the ellipse E given by x^2 18 + y^2 12 = 1 . Let H be the hyperbola whose eccentricity is the reciprocal of the eccentricity of E and whose foci are the same as that of E 2026Passage: Consider the ellipses given by x^2 + 4y^2 = 1 and 4x^2 + y^2 = 1 . Question: Let P be the point in the first quadrant where the given ellipses intersect. If is the acute a 2026Let x^2 f(a^2+7a+3) + y^2 f(3a+15) = 1 represent an ellipse with major axis along y -axis, where f is a strictly decreasing positive function on R . If the set of all possible valu 2026The eccentricity of an ellipse E with centre at the origin O is 3 2 and its directrices are x = 4 6 3 . Let H: x^2 a^2 - y^2 b^2 = 1 be a hyperbola whose eccentricity is equal to t 2026Let x = 9 be a directrix of an ellipse E , whose centre is at the origin and eccentricity is 1 3 . Let P( , 0) , > 0 , be a focus of E and AB be a chord passing through P . Then th 2026Let a focus of the ellipse E: x^2 a^2 + y^2 b^2 = 1 be S(4, 0) and its eccentricity be 4 5 . If the point P(3, ) lies on E and O is the origin, then the area of POS is equal to: 2026Let A be the point (3, 0) and circles with variable diameter AB touch the circle x^2 + y^2 = 36 internally. Let the curve C be the locus of the point B. If the eccentricity of C is 2026Let an ellipse x^2 a^2 + y^2 b^2 = 1 , a < b , pass through the point (4, 3) and have eccentricity 5 3 . Then the length of its latus rectum is : 2026 Full Ellipse list All JEE Advanced PYQs