QQuestionMathematics
QuestionMathematics
In the $\boldsymbol{x y}$-plane, a line with equation $\mathbf{2 y}=\mathbf{4 . 5}$ intersects a parabola at exactly one point. If the parabola has equation $\mathbf{y}=-\mathbf{4 x ^ { 2 } }+\mathbf{b x}$, where $\mathbf{b}$ is a positive constant, what is the value of $\mathbf{b}$ ?
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Answer
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Step 1:: First, we need to find the point of intersection between the line and the parabola.
y = \frac{4.5}{2} = 2.25
Step 2:: Now, we will substitute this value of $y$ into the equation of the parabola, $y = - 4x^2 + bx$, to find the corresponding value of $x$:
2.25 = -4x^2 + bx
Step 3:: To solve for $x$, we first rewrite the equation in standard form by bringing all terms to the left side:
4x^2 - bx + 2.25 = 0
Step 4:: Since the parabola intersects the line at exactly one point, this quadratic equation has only one real root.
b^2 - 4(4)(2.25) = 0
Step 5:: Expanding and simplifying the equation above, we get:
b^2 = 36
Step 6:: Taking the square root of both sides, we find the value of $b$:
b = 6
Final Answer
The value of $b$ is 6.
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