Question

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

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

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).

Key concepts

Common Chord

In the world of parabolas and geometry, a **common chord** is quite an interesting concept. A chord, in general, is a straight line segment that joins two points on a curve. When discussing parabolas, a common chord is the line segment that connects the points where two parabolas intersect. These intersecting points lie on both curves simultaneously, making the chord common to both. This overlap is what gives the truncated portion between these points its name. Understanding the common chord is crucial because it represents symmetry and relationships between the shapes of parabolas and their overlapping regions.

System of Equations

The method to identify where two parabolas meet involves setting up a **system of equations**. A system of equations is essentially a set of equations with multiple variables that you solve together. For the parabolas given by the equations

• \( y^2 = x \)
• \( x^2 = y \)

The objective is to find values for x and y that satisfy both equations at the same time. By substituting one equation into the other, you can reduce the problem to a simpler form, isolating either variable. This step helps in determining specific points of intersection, which are key for defining the common chord's length and position.

Distance Formula

The **distance formula** is a powerful tool for calculating the length between two points in a plane. This formula is especially useful after determining the points of intersection between parabolas. The formula is:\[ \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2} \]By inserting the coordinates of the points, you can find the length of the segment connecting them. In the case of the parabolas

• \((0,0)\)
• \((1,1)\)

represented the intersection points. Plugging these into the formula yields \( \sqrt{2} \), giving us the length of the common chord between these parabolas. Using this formula allows for precise calculations fundamental in geometry.

Real Intersection Points

When solving for where parabolas meet, **real intersection points** are crucial. These points are the coordinates at which both parabolas cross each other creating a space of interaction. Using algebra to work through the system of equations involving our two parabolas, imaginary solutions, though mathematically interesting, do not apply to our physical intersection points. For this problem, after factoring and solving, we identified the real solutions.

• \( y = 0 \) leading to the intersection point \((0, 0)\)
• \( y = 1 \) leading to the intersection point \((1, 1)\)

These real points were essential for determining the actual length of the common chord. Grokking these concepts helps in visually and algebraically understanding how shapes intersect and affect each other in a coordinate system.