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Chapter 15: Problem 1386

The length of the common chord of the parabolas \(\mathrm{y}^{2}=\mathrm{x}\) and \(\mathrm{x}^{2}=\mathrm{y}\) is (a) 1 (b) \(\sqrt{2}\) (c) \(4 \sqrt{2}\) (d) \(2 \sqrt{2}\)

Short Answer

Expert verified
The length of the common chord of the parabolas \(y^2=x\) and \(x^2=y\) is \(\sqrt{2}\), which corresponds to option (b).

Step by step solution

01

In order to find the intersection points, we need to solve the system of equations formed by the parabolas: \[\begin{cases} y^2 = x \\ x^2 = y \end{cases}\] We can substitute the value of x from the equation \(y^2 = x\) into the equation \(x^2 = y\) and get: \[ (y^2)^2 = y \] \[ y^4-y = 0 \] We are trying to find the common real roots of both equations for y. We can factor y out of the equation above: \[ y(y^3-1)=0 \] \[ y(y-1)(y^2+y+1)=0 \] This equation has roots y=0, y=1, and two complex roots when \(y^2+y+1=0\). Therefore, we have two intersection points with real coordinate values: \[(0,0)\] from y=0, and \[(1,1)\] from y=1. #Step 2: Calculate the length of the common chord

Now that we have the coordinates of the intersection points, we can use the distance formula to find the length of the common chord: \[ length = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2} \] Plugging in the coordinates of our intersection points: \[ length = \sqrt{(1 - 0)^2 + (1 - 0)^2} \] \[ length = \sqrt{2} \] Thus, the length of the common chord of the parabolas is \(\sqrt{2}\), which corresponds to option (b).

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