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Three points lie on the graph of the parabola $y = x^2$.  The three points form an equilateral triangle.  One of the vertices is at the vertex of the parabola, and the other two points lie on the line $y = k$.  What is the value of $k?$

 Aug 25, 2023
 #1
avatar+126978 
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We can find the intersection of the  line   y = sqrt (3)x  and the  parabola to  give  us the answer

 

x^2 = sqrt (3)x

 

x^2 - sqrt (3)x  = 0

 

x ( x - sqrt 3)  = 0

 

Setting the second  factor to  0  and solving for x  produces the  x coordinate of  one of the  intersections = sqrt 3

 

The y coordinate is  y= x^2 =   (sqrt 3)^2 =  3 =  k

 

 

cool cool cool

 Aug 25, 2023

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