Solution to Braintease No. 1

SOLUTION TO BRAINTEASER NO. 1 DISCLAIMER This solution is but one of the many possible methods for solving this problem. Various other solutions may lead to the same answers. This specific solution was provided by one of the members of the University of the Philippines Civil Engineering Society.

PROBLEM:

A circle with center at (0,a) intersects the parabola y = x2 at the points (2,4) and (-2,4). Find a.

SOLUTION: Looking closely, we are sure you have realized that the problem requires a limiting condition since the circle’s radius and the ordinate a could have an infinite number of values. Failing to define this limiting condition was a fault on our part. However, for the sake of solving this problem, here is a method to solving the problem in its simplest form – when the circle tangentially intersects the parabola at the given points: If the parabola is tangential to the circle at the given points, then the slope of the tangent lines of the parabola and the circle should be equal. Furthermore, we can generate the equation of the parabola to be y = x2 and the equation of the circle to be x2 + (y - a)2 = r2, where r is a constant. Differentiate y = x2 in terms of x: 𝑑𝑦 𝑑𝑥

(1)

= 2𝑥

Use implicit differentiation for the circle’s equation x2 + (y – a)2 = r2: 𝑑𝑦 𝑑𝑥

𝑥

(2)

= − (𝑦−𝑎)

𝐸𝑞𝑢𝑎𝑡𝑒 𝑒𝑞𝑢𝑎𝑡𝑖𝑜𝑛𝑠 (1) 𝑎𝑛𝑑 (2): 𝑥 2𝑥 = − (𝑦−𝑎) (3) 𝑆𝑢𝑏𝑠𝑡𝑖𝑡𝑢𝑡𝑒 𝑡ℎ𝑒 𝑐𝑜𝑜𝑟𝑑𝑖𝑛𝑎𝑡𝑒𝑠 𝑜𝑓 𝑜𝑛𝑒 𝑜𝑓 𝑡ℎ𝑒 𝑝𝑜𝑖𝑛𝑡𝑠 𝑖𝑛𝑡𝑜 𝑒𝑞𝑢𝑎𝑡𝑖𝑜𝑛 (3): (2)

2(2) = − ((4)−𝑎)

or

(−2)

2(−2) = − ((4)−𝑎)

Solve for a.

ANSWER:

a = 9/2

UNIVERSITY OF THE PHILIPPINES CIVIL ENGINEERING SOCIETY Department of Civil Engineering College of Engineering and Agro-industrial Technology University of the Philippines Los Baños