Quantrex Quantrex AcademyJEE · NEET · NDA PYQs with solutions Open app
JEE Advanced2008MathematicsStraight LinesActual

A straight line through the vertex P of a P Q R intersects the side Q R at the point S and the circumcircle of the P Q R at the point T . If S is not the centre of the circumcircle, then

Options

  1. A1 P S + 1 S T < 2 Q S S R
  2. B1 P S + 1 S T > 2 Q S S R
  3. C1 P S + 1 S T < 4 Q R
  4. D1 P S + 1 S T > 4 Q R

Correct answer

B. 1 P S + 1 S T > 2 Q S S R

Step-by-step solution

Let a straight line through the vertex P of a given P Q R intersects the side Q R at the point S and the circumcircle of P Q R at the point T . Points P, Q, R, T are concyclic, hence array rlrl P S S T & =Q S S R & & P₁ & 2 P S S T & & Now, P S+S T 2 & P S S T & [ AM GM ] array and 1 P S + 1 S T 2 P S S T = 2 Q S S R Also, S Q+Q R 2 S Q S R Q R 2 S Q S R array ll & 1 S Q S R 2 Q R 2 S Q S R 4 Q R Hence, & 1 P S + 1 S T 2 Q S S R 4 Q R . array

Practice Straight Lines on Quantrex Academy →

More from Straight Lines

If a straight line drawn through the point of intersection of the lines 4x+3y-1=0 and 3x+4y-1=0 , meets the co-ordinate axes at the points P and Q, then the locus of the mid point 2026From the point (-1, -1) , two rays are sent making angles of 45° with the line x + y = 0 . These rays get reflected from the mirror x + 2y = 1 . If the equations of the reflected r 2026In an equilateral triangle PQR , let the vertex P be at (3, 5) and the side QR be along the line x + y = 4 . If the orthocentre of the triangle PQR is ( , ) , then 9( + ) is equal 2026Let P(3 , 2 ) , 0 , be a point on the ellipse x^2 9 + y^2 4 =1 , Q be a point on the circle x^2+y^2-14x-14y+82=0 and R be a point on the line x+y=5 such that the centroid of the tr 2026Let A,B be points on the two half-lines x- 3 |y|= , >0 at a distance of from their point of intersection P . The line segment AB meets the angle bisector of the given half-lines at 2026Let the line L₁ : x + 3 = 0 intersect the lines L₂ : x - y = 0 and L₃ : 3x + y = 0 at the points A and B , respectively. Let the bisector of the obtuse angle between the lines L₂ a 2026Let the vertex A of a triangle ABC be (1, 2) , and the mid-point of the side AB be (5, -1) . If the centroid of this triangle is (3, 4) and its circumcenter is ( , ) , then 21( + ) 2026Let the mid points of the sides of a triangle ABC be ( 5 2 , 7 ) , ( 5 2 , 3 ) and (4, 5) . If its incentre is (h, k) , then 3h + k is equal to : 2026 Full Straight Lines list All JEE Advanced PYQs