The area in square units bounded by the normal at (1,2) to the parabola y²=4 x, X -axis and the curve is given by

Step-by-step solution

We have, y²=4 x Differentiating w.r.t. x , we get aligned &2 y d y d x =4 d y d x = 2 y & ( d y d x )_ (1,2) = 2 2 =1 aligned Equation of normal to the curve at (1,2) is aligned & aligned y-y₁=& 1 (- d y d x ) (x-x₁ ) (y-2)=- 1 1 (x-1) aligned & y-2=-x+1 x+y=3 & The line x+y=3 meets the X-axis at x=3 & Required area = ₀¹ 4 x d x+ ₁³(3-x) d x &=2 [ x^ 3 / 2 3 / 2 ]₀¹+ [3 x- x² 2 ]₁³ &= 4 3 (1)+ [9- 9 2 -3+ 1 2 ]= 4 3 + ( 9 2 - 5 2 ) &= 4 3 + 4 2 = 4 3 +2= 10 3 sq units aligned