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JEE Main 20254 AprilEvening Shift
Exam
JEE Main
Year
2025
Date
4 April
Shift
Evening Shift
Subject
Mathematics
Chapter
Parabola

JEE Main 2025 · 4 April · Evening Shift · Mathematics · Parabola

A line passing through the point A(-2,0) , touches the parabola P:...

A line passing through the point A(−2,0)\mathrm{A}(-2,0), touches the parabola P:y2=x−2P: y^2=x-2 at the point BB in the first quadrant. The area, of the region bounded by the line AB , parabola P and the x -axis, is :-

Options

  1. A73\frac{7}{3}
  2. B22
  3. C83\frac{8}{3}
  4. D33

Correct answer

C. 83\frac{8}{3}

Step-by-step solution

Tangenty=m(x+2)y2=x−2(m(n+2))2=n−2m2x2+(4m2−1)x+(4m2+2)=0D=0(4m2−1)2−4m2(4m2+2)=0m=14y=14(n+2)\begin{aligned}& y=m(x+2) \\ & y^2=x-2 \\ & (m(n+2))^2=n-2 \\ & m^2 x^2+\left(4 m^2-1\right) x+\left(4 m^2+2\right)=0 \\ & D=0 \\ & \left(4 m^2-1\right)^2-4 m^2\left(4 m^2+2\right)=0 \\ & m=\frac{1}{4} \\ & y=\frac{1}{4}(n+2)\end{aligned}and point of tangency (6,2)(6,2) Area A=∫02((y2+2)−(4y−2))dyA=83\begin{aligned}& \text{ Area } A=\int_0^2\left(\left(y^2+2\right)-(4 y-2)\right) d y \\ & A=\frac{8}{3}\end{aligned}$$option (3)
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