Quantrex Quantrex AcademyJEE · NEET · NDA PYQs with solutions Open app
JEE Main20249 Apr 2024Evening ShiftMathematicsParabolaActual

Let A, B and C be three points on the parabola y^2=6 x and let the line segment A B meet the line L through C parallel to the x -axis at the point D . Let M and N respectively be the feet of the perpendiculars from A and B on L . Then ( A M B N C D )^2 is equal to _________

Correct answer

0

Step-by-step solution

aligned & m _ AB = m _ AD & 2 t ₁+ t ₂ = 2 a ( t ₁- t ₃ ) at ₁^2- & at ₁^2- = a t ₁^2- t ₁ t ₃+ t ₁ t ₂- t ₂ t ₃ & = a ( t ₁ t ₃+ t ₂ t ₃- t ₁ t ₂ ) & AM = |2 a ( t ₁- t ₃ ) |, BN = |2 a ( t ₂- t ₃ ) |, & CD = | at ₃^2- | aligned aligned & CD = | at ₃^2- a ( t ₁ t ₃+ t ₂ t ₃- t ₁ t ₂ ) | &= a | t ₃^2- t ₁ t ₃- t ₂ t ₃+ t ₁ t ₂ | &= a | t ₃ ( t ₃- t ₁ )- t ₂ ( t ₃- t ₁ ) | & CD = a | ( t ₃- t ₂ ) ( t ₃- t ₁ ) | & ( AM BN CD )^2= 2 a ( t ₁- t ₃ ) 2 a ( t ₂- t ₃ ) a ( t ₃- t ₂ ) ( t ₃- t ₁ ) ^2 & 16 a ^2=16 9 4 =36 al

Practice Parabola on Quantrex Academy →

More from Parabola

Let T be the tangent to the parabola y^2 = 16x at the point (64, 32) . Let L be the tangent to the same parabola at another point (x₁, y₁) on the parabola. If L and T are perpendic 2026Let O be the vertex of the parabola y^2=4x and its chords OP and OQ are perpendicular to each other. If the locus of the mid-point of the line segment PQ is a conic C, then the len 2026Let chord PQ of length 3 13 of the parabola y^2 = 12x be such that the ordinates of points P and Q are in the ratio 1:2 . If the chord PQ subtends an angle at the focus of the para 2026Let the point P be the vertex of the parabola y = x^2 - 6x + 12 . If a line passing through the point P intersects the circle x^2 + y^2 - 2x - 4y + 3 = 0 at the points R and S , th 2026Let the directrix of the parabola P: y^2 = 8x , cut x -axis at the point A . Let B( , ) , > 1 , be a point on P such that the slope of AB is 3/5 . If BC is a focal chord of P , the 2026Let A,B and C be the vertices of a variable right angled triangle inscribed in the parabola y^2=16x . Let the vertex B containing the right angle be (4,8) and the locus of the cent 2026Consider the parabola P : y^2 = 4kx and the ellipse E : x^2 a^2 + y^2 b^2 = 1 . Let the line segment joining the points of intersection of P and E , be their latus rectums. If the 2026Let the parabola y = x^2 + px + q passing through the point (1, -1) be such that the distance between its vertex and the x -axis is minimum. Then the value of p^2 + q^2 is: 2026 Full Parabola list All JEE Main PYQs