solve---**Question 2:**
For which value of c is the line y = 3x + c a tangent to the parabola with equation y = x² - 5x + 7?
**Question 6:**
Consider the curve y = x² + 4x - 1 and the line y = 2x + c. Find the values of c for which the line:
a meets the curve twice
b is a tangent to the curve
c does not meet the curve.
**Chart/Diagram Description (associated with Question 6):**
Type: Coordinate plane graph.
Main Elements:
- X-axis and Y-axis are shown, labeled "x" and "y" respectively, with arrows indicating the positive direction. The origin is marked with a circle.
- A purple curve representing a parabola is drawn. It opens upwards and its vertex is in the third quadrant. It is labeled with the equation "y = x² + 4x - 1".
- A green line is drawn with a positive slope. It intersects the Y-axis at a point labeled "c". It is labeled with the equation "y = 2x + c".
- In the depiction, the line intersects the parabola at two distinct points.
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Today we'll explore how to find when a line is tangent to a parabola. A tangent line touches the curve at exactly one point. We'll use the discriminant method to solve these problems systematically.
Let's solve Question 2. We need to find the value of c that makes the line y equals 3x plus c tangent to the parabola y equals x squared minus 5x plus 7. First, we set the equations equal. Then rearrange to standard form. For tangency, the discriminant must equal zero. Calculating gives us 64 minus 28 plus 4c equals zero, so c equals negative 9.
Now let's tackle Question 6. We have the curve y equals x squared plus 4x minus 1 and the line y equals 2x plus c. To find when they intersect, we set the equations equal and rearrange to get x squared plus 2x minus quantity 1 plus c equals zero. The discriminant is 8 plus 4c. This discriminant will determine how many intersection points we have.
Now we analyze the three cases using the discriminant. When delta is greater than zero, meaning c is greater than negative 2, the line meets the curve twice. When delta equals zero, meaning c equals negative 2, the line is tangent to the curve. When delta is less than zero, meaning c is less than negative 2, the line doesn't meet the curve at all. Let's see this visually as we change the value of c.
Let's summarize our solutions. For Question 2, we found that c equals negative 9 makes the line y equals 3x plus c tangent to the parabola. For Question 6, we determined that when c is greater than negative 2, the line meets the curve twice. When c equals negative 2, the line is tangent. When c is less than negative 2, there's no intersection. The key method is using the discriminant: positive for two intersections, zero for tangency, and negative for no intersections.