Question

instructions
choose the best answer. if necessary, use the paper you were given.
question
in the figure above, the parabola is the graph of ( y = x ^ { 2 } ). if the y - coordinate of p and the y - coordinate of q are each 6, what is the length of ( overline { p q } ) ?
12
6
( 2 sqrt { 6 } )
( sqrt { 6 } )

Explanation:

Step1: Find x - coordinates of P and Q

Since \(y = x^{2}\) and \(y = 6\), then \(x^{2}=6\). Solving for \(x\), we get \(x=\pm\sqrt{6}\). Let the coordinates of \(P\) be \((-\sqrt{6},6)\) and of \(Q\) be \((\sqrt{6},6)\).

Step2: Calculate the length of \(PQ\)

The distance formula between two points \((x_1,y_1)\) and \((x_2,y_2)\) is \(d=\sqrt{(x_2 - x_1)^{2}+(y_2 - y_1)^{2}}\). Here, \(y_1 = y_2=6\), so \(d=\sqrt{(\sqrt{6}-(-\sqrt{6}))^{2}+(6 - 6)^{2}}=\sqrt{(2\sqrt{6})^{2}} = 2\sqrt{6}\).

Answer:

C. \(2\sqrt{6}\)