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Minima/ maxima (statement questions) - Geometry questions
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Minima/ maxima (statement questions) - Geometry questions
Last updated at April 19, 2021 by Teachoo
Example 41 An Apache helicopter of enemy is flying along the curve given by π¦= π₯^2 + 7. A soldier, placed at (3, 7), wants to shoot down the helicopter when it is nearest to him. Find the nearest distance.Given curve y = x2 + 7 Let (π₯,π¦) be any point on parabola π¦=π₯2+7 Let D be required Distance between (π₯,π¦) & (3 , 7) D = β((πβπ)^π+(π βπ)^π ) = β(9+π₯^2β6π₯+49+π¦^2β14π¦) = β(π₯^2+π¦^2β6π₯β14π¦+58) Since point (π₯ , π¦) is on the parabola π¦=π₯2+7 (π , π) will satisfy the equation of parabola Putting π₯ and π¦ in equation π=π^π+π Putting value of π¦=π₯^2+7 D = β(π₯^2+π¦^2β6π₯β14π¦+58) D = β(π₯^2+γ(π₯^2+7)γ^2 β 6π₯β14(π₯^2+7)+58) D = β(π₯^2+π₯^4+49+14π₯^2 β 6π₯β14π₯^2β98+58) D = β(π^π+π^π βππ+π) We need to minimize D, but D has a square root Which will be difficult to differentiate Let Z = D2 Z = π^π+π^π βππ+π Since D is positive, D is minimum if D2 is minimum So, we minimize Z = D2 Differentiating Z Z =π₯^4+π₯^2 β6π₯+9 Differentiating w.r.t. h Zβ = π(π₯^4 + π₯^2 β 6π₯ + 9)/πβ Zβ = 4π₯^3+2π₯ β6 Putting Zβ = 0 4π₯^3+2π₯ β6=0 Factorizing Zβ Zβ(1) = 4(1)3 + 2(1) β 6 = 4 + 2 β 6 = 0 Hence, (x β 1) is a factor of 4x3 β 2x β 6 Thus, 4π₯^3+2π₯ β6=0 (π₯β1)(4π₯^2+4π₯+6)=0 2x2 + 2x + 3 = 0 x = (β2 Β± β(4 β 4(2)(3)))/4 = (β2 Β± β(βππ))/4 This is not possible as there are no real roots. Checking sign of π^β²β² " " ππ/ππ₯=4π₯^3+2π₯ β6 Differentiating again w.r.t x (π^2 π)/(ππ₯^2 )=4 Γ 3π₯^2+2 (π^2 π)/(ππ₯^2 )=12π₯^2+2 Since π^β²β² > 0 for x = 1 β΄ Z is minimum when x = 1 Thus, D is Minimum at x = 1 Finding Minimum value of D D = β(π^π+π^π βππ+π) Putting x = 1 D = β(1^4+1^2β6(1)+9) D = βπ Hence, shortest distance isβπ