In the x y -plane, a line with equation 2 y = 4.5 intersects a parabola at exactly one point.

Question

In the x y -plane, a line with equation 2 y = 4.5 intersects a parabola at exactly one point. If the parabola has equation y = - 4 x 2 + b x , where b is a positive constant, what is the value of b ?

Accepted Answer

The correct answer is 6 . It&rsquo;s given that a line with equation 2 y = 4 . 5 intersects a parabola with equation y = - 4 x 2 + b x , where b is a positive constant, at exactly one point in the xy -plane. It follows that the system of equations consisting of 2 y = 4 . 5 and y = - 4 x 2 + b x has exactly one solution. Dividing both sides of the equation of the line by 2 yields y = 2 . 25 . Substituting 2.25 for y in the equation of the parabola yields 2 . 25 = - 4 x 2 + b x . Adding 4 x 2 and subtracting b x from both sides of this equation yields 4 x 2 - b x + 2 . 25 = 0 . A quadratic equation in the form of a x 2 + b x + c = 0 , where a , b , and c are constants, has exactly one solution when the discriminant, b 2 - 4 a c , is equal to zero. Substituting 4 for a and 2.25 for c in the expression b 2 - 4 a c and setting this expression equal to 0 yields b 2 - 4 ( 4 ) ( 2 . 25 ) = 0 , or b 2 - 36 = 0 . Adding 36 to each side of this equation yields b 2 = 36 . Taking the square root of each side of this equation yields b = &#177; 6 . It&rsquo;s given that b is positive, so the value of b is 6 .

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