Aiming the Communication Device You reach the communication device and are alerted by your computer that the communication device is 2 surrounded by a force field that can be described by the equation, :n2 + y2 = (% :1:2 + % y2 y) , with the origin being where you located this device. The inhabitants of this planet did not want unwelcomed guests, so they made sure to guard the use of this device. Your computer helpfully generates a plot of this equation and tells you that it is a cardioid (whatever that means). The communication device can be taken to the very edge of the force field, but not beyond it. In fact, the only way to use the device is to align it tangent to the force field. Your computer also recovers some files that give points along the force field where the communication device can be used to communicate with different planets. Communication with your home planet can be achieved from the point on the force field, (6, 0). If you provide your computer with the equation of the line that satisfies these conditions, you will be able to send a message to your home world. What equation do you provide your computer? To determine the equation of the line that is tangent to the cardioid at the point (6, 0), we need to first confirm the form of the cardioid given by the equation: Step 1: Simplify the given equation Rewrite the equation: _1 ) We can let = 5, then the equation becomes: 2 +y? = (az? + ay? y)2 Step 2: Derive the general form of a cardioid A standard cardioid equation in polar coordinates is: r = a(l + cos0) However, the given equation is in Cartesian coordinates. To convert it, we generally use: r =7rcosb y = rsinf Step 3: Find the point of tangency Given the point (6, 0), which lies on the cardioid, and the need for a tangent line at this point, we proceed by taking the derivative of the cardioid to find the slope of the tangent line at this point. Step 4: Compute the slope To find the slope, we differentiate implicitly: At the point (6, 0): 1 1 1 = [=. [} . [12)1 6+0 (6 36+60 0)(6( ) ) 6=6(21) 6 =6 d1 Hence, we need to find the slope ar by further simplification and differentiating the cardioid equation properly. Step 5: Write the equation of the tangent line Assuming we find the correct slope m from the above differentiation, the tangent line at point (6, 0) would have the form: y=m(z6) Simplified Answer: To make the solution straightforward for communication: If we know the tangent at (6, 0) directly: For cardioid equations, typically tangents at horizontal intercepts (like (6, 0)) are vertical, thus: T=0 However, if we find the slope m and derive correctly, it would take the final form of = mz + b, Given it is a vertical line for the cardioid at horizontal intercept, the answer simplifies to: w=6 So, the equation provided to the computer is: =