Consider a circle x 2 + ( y − 1 ) 2 = 1 and the parabola y = − 4 x 2 . The common tangents to the two curves constitute a triangle ABC, with points A and B on the x-axis and C on the y-axis. CA prod

Question

Consider a circle x 2 + ( y − 1 ) 2 = 1 and the parabola y = − 4 x 2 ​ . The common tangents to the two curves constitute a triangle ABC, with points A and B on the x-axis and C on the y-axis. CA produced touches the parabola at P, and CB produced touches the parabola at Q. (a) Find the equation of the common tangent BC. (b) Find the area of the portion between the upper arc of the circle and the common tangents QC and PC. (c) Find the area enclosed by the parabola y = − 4 x 2 ​ , the x-axis, and the lines AP and BQ.

TestHub · Accepted Answer

x 2 + ( y − 1 ) 2 = 1 x 2 = − 4 y . ​ Equation of tangent of (ii) y = mx + m 2 equation (iii) touching circle (i) Corrected complete working: Tangent: y = m x + m 2 . m 2 + 1 ​ ∣ m 2 − 1∣ ​ = 1 m 2 ( m 2 − 3 ) = 0. ​ m = 0 : y = 0. m = ± 3 ​ : y = ± 3 ​ x + 3. BC : y = 3 ​ x + 3. ​ S = ( − 3 ​ /2 , 3/2 ) , R = ( 3 ​ /2 , 3/2 ) , P = ( 2 3 ​ , − 3 ) . ​ Upper area: 2 ∫ − 3 ​ /2 0 ​ ( 3 ​ x + 2 − 1 − x 2 ​ ) d x = 3 ​ − π /3. ​ Lower area = 2 ( I 1 ​ − I 2 ​ ) , I 1 ​ = ∫ 3 ​ 2 3 ​ ​ ( − 3 ​ x + 3 ) d x = − 3 3 ​ /2 , I 2 ​ = ∫ 0 2 3 ​ ​ ( − x 2 /4 ) d x = − 2 3 ​ . 2 ( I 1 ​ − I 2 ​ ) = 3 ​ . ​

Mathematics - Area Under Curves Question with Solution | TestHub

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