Porisma

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Optimise and differentiate a triangle on an implicit curve

Point Q(x,y)Q(x,y) lies on 4x2+xy+y2=364x^2+xy+y^2=36, with y≥0y\ge0. The curve meets the x-axis at P,RP,R as drawn.

(i) Show that triangle PQRPQR has area A=3yA=3y. marks

(ii) Find dy/dxdy/dx in terms of x,yx,y. marks

(iii) Hence find the x-value where AA is stationary and use the second derivative test to determine its nature. marks

If xx increases at 88 units/s, find the rate of change of AA when x=0x=0. marks

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Upper part, y≥0, of the closed curve 4x²+xy+y²=36, with x-axis intersections P and R and a general point Q(x,y) on the upper-left arc. No numerical intercepts are supplied in this given diagram.
Upper part, y≥0, of the closed curve 4x²+xy+y²=36, with x-axis intersections P and R and a general point Q(x,y) on the upper-left arc. No numerical intercepts are supplied in this given diagram.
Upper part, y≥0, of the closed curve 4x²+xy+y²=36, with x-axis intersections P and R and a general point Q(x,y) on the upper-left arc. No numerical intercepts are supplied in this given diagram.

[2013/DHS/I/11]

CJC · Review problems · Question 9

The base is fixed at the two axis intersections; only the height changes.

Read the worked solution