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Most Important Selected Qs for JEE AdvancedMathematicsParabola

Let two parabolas be S₁: y^2=4 a x and S₂: y^2=-4 a x . From any point P on S₁ , tangents are drawn to S₂ touching it at Q and R , then:

Options

  1. Aline Q R is tangent to S₁
  2. Bline Q R neither touches nor intersect S₁
  3. Cif normal at any point A(t) on S₁ is tangent to S₂ then t^2= 2 -1
  4. Dif normal at any point A(t) on S₁ is tangent to S₂ then t^2= 2 +1

Correct answer

C. if normal at any point A(t) on S₁ is tangent to S₂ then t^2= 2 -1

Step-by-step solution

Equation of chord of contact Q R from P (a t^2, 2 a t ) will be y 2 a t=-2 a (x+a t^2 ) y(-2 a t)=2 a (x+a t^2 ) which is tangent to parabola at y^2=4 a x at (a t^2,-2 a t ) and normal at A(t) on S₁ y+t x=2 a t+a t^3 y=-t x+2 a t+a t^3 ...(1) Tangent at S₂ will be : y=m x- a m ...(2) aligned & m=-t and 2 a t+a t^3=- a m & 2 a t+a t^3= a t t^4+2 t^2=1 (t^2+1 )^2=2 aligned

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