Intersecting lines

Geometry Level 3

If ( a , b ) (a,b) is a point of intersection of the lines x cos ⁡ θ + y sin ⁡ θ = 3 x\cos\theta+y\sin\theta=3 and x sin ⁡ θ − y cos ⁡ θ = 4 x\sin\theta -y\cos\theta=4 , where θ \theta is a parameter, then find the maximum value of 2 W 2^W where W = ( a + b 2 ) W = {\left( \dfrac{a+b}{\sqrt{2}} \right) } .

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128 64 32 None of the given values. 16

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Sandeep Bhardwaj by Sandeep Bhardwaj , 28, India

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1 solution

Prakhar Gupta
May 1, 2015

We have to first of all find the intersection point of the two lines. x . cos ⁡ θ + y . sin ⁡ θ = 3 x.\cos\theta + y.\sin\theta = 3 Multiply by cos ⁡ θ \cos\theta . x . cos ⁡ 2 θ + y . sin ⁡ θ . cos ⁡ θ = 3. cos ⁡ θ x.\cos^{2}\theta + y.\sin\theta.\cos\theta = 3.\cos\theta Another equation is :- x . sin ⁡ θ − y . cos ⁡ θ = 4 x.\sin\theta - y.\cos\theta = 4 Multiply by sin ⁡ θ \sin\theta . x . sin ⁡ 2 θ − y . sin ⁡ θ . cos ⁡ θ = 4. sin ⁡ θ x.\sin^{2}\theta - y.\sin\theta.\cos\theta = 4.\sin\theta Adding the two equation, we get:- x = 3 cos ⁡ θ + 4 sin ⁡ θ x = 3\cos\theta + 4\sin \theta Similarly we can find y y . y = 3 sin ⁡ θ − y cos ⁡ θ y = 3\sin\theta - y\cos\theta Hence a + b = 7 sin ⁡ θ − cos ⁡ θ a +b = 7\sin\theta - \cos\theta To find the maximum value of the given expression, we need to find the maximum value of a + b a+b . M a x ( a + b ) = 7 2 + 1 2 = 50 = 5 2 Max(a+b) = \sqrt{7^{2}+1^{2}}=\sqrt{50} = 5\sqrt{2} Hence the required value is 2 5 = 32 \boxed{2^{5} = 32}

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Thank you for your solution.

I don't understand how you can so quickly find that the maximum of 7 sin ⁡ θ − cos ⁡ θ 7\sin\theta - \cos\theta is 7 2 + 1 2 \sqrt{7^2+1^2} , though. I can only do it with calculus. What's your insight here?

Laurent Shorts - 5 years, 2 months ago

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