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

Let S be the focus and D be the directrix of the parabola x^2 = 12y . A point A lies on the parabola such that the tangent to the parabola at A makes an angle of 3 with the positive x-axis. A point B lies on the directrix D such that SA AB . If the area of SAB is K 3 , then the value of K is

Options

  1. A48
  2. B96
  3. C16
  4. D12

Correct answer

A. 48

Step-by-step solution

The equation of the parabola is x^2 = 12y . Differentiating with respect to x , we get 2x = 12 dy dx dy dx = x 6 . The tangent at A makes an angle of 3 with the positive x-axis, so its slope is ( 3 ) = 3 . Equating the slopes: x 6 = 3 x = 6 3 . Substituting x into the parabola's equation: y = (6 3 )^2 12 = 108 12 = 9 . Thus, A is (6 3 , 9) . For x^2 = 12y , 4a = 12 a = 3 . The focus is S(0, 3) and the directrix D is y = -3 . The slope of SA is m_ SA = 9 - 3 6 3 - 0 = 6 6 3 = 1 3 . Since SA AB , the slope of AB is m

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