Math Problem Statement
Solution
Let's break down and solve the problem step by step from the image:
Given:
The equation of the curve is: where is a constant.
The line given is: which is tangent to the curve. We are tasked with finding the value of .
Steps to solve:
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Set the two equations equal to each other: Since the line is tangent to the curve, their equations will intersect at exactly one point. Set the equations equal:
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Simplify to quadratic form: Rearranging the equation: Simplify the terms: This is a quadratic equation in the form .
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Set the discriminant equal to zero: Since the line is tangent, the discriminant of the quadratic equation must be zero for the quadratic to have exactly one solution. The discriminant is given by: For the quadratic equation , the coefficients are: Setting the discriminant to zero:
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Solve the discriminant for : Expanding both terms: Expand the second term: Now substitute into the discriminant equation: Simplify: Divide through by -7: Factorize: So, .
Thus, the value of is 3.
Would you like any further details or explanations on any part of this solution?
Here are 5 related questions to expand understanding:
- What happens if the line is not tangent but intersects the curve at two points?
- How do you calculate the discriminant of a quadratic equation in general?
- What is the geometric significance of setting the discriminant equal to zero?
- How do different values of affect the shape of the parabola given by the equation of the curve?
- What if the curve was a cubic function? How would you find the value of ?
Tip:
Always check the discriminant when dealing with tangency conditions, as it provides key insights into the number of intersections.
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Math Problem Analysis
Mathematical Concepts
Algebra
Quadratic Equations
Discriminant
Tangency Conditions
Formulas
Quadratic equation: ax^2 + bx + c = 0
Discriminant: Δ = b^2 - 4ac
Theorems
Discriminant theorem for quadratic equations
Suitable Grade Level
High School (Grades 10-12)
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