91Ó°ÊÓ

Understanding common tangents is key in solving problems about curves with shared tangents. A common tangent is a single line that touches two curves at distinct points but does not cut across them. In the given exercise, the two curves, \( y = x^2 + ax + b \) and \( y = cx - x^2 \), share a common tangent at the point \((1,0)\). This means that the tangent line at this point has the same slope for both curves, forming an intersection. The tangent acts as a bridge that both curves touch, which allows us to equate their slopes through derivative calculus. Solving for shared tangents typically requires evaluating the curves at the common point and ensuring the derivatives are equal at this contact point, giving rise to equations that can be solved for the unknowns.

The derivative is a mathematical tool that helps us find the rate of change of a function. It's like asking how steep a hill is at any particular point on the function's curve. In our exercise, we use derivatives to calculate the slope of the tangent line at a specific point on each curve. For each of our curves, we take the derivative:

• For \( y = x^2 + ax + b \), the derivative is \( \frac{dy}{dx} = 2x + a \).
• For \( y = cx - x^2 \), it is \( \frac{dy}{dx} = c - 2x \).

By differentiating, we find these expressions which give us the slope at any given \( x \). By equating the derivatives at the point \((1,0)\), we ensure that the slopes of both curves are identical, signifying the curves share the same tangent. This step is crucial in solving for the parameters \( a \), \( b \), and \( c \).

The slope of a curve describes its steepness at any given point. You can think of it as a snapshot of the curve's inclination, just like how steep a road seems when you're driving on it at that instant. In the context of our exercise, this slope is determined by the derivative. For example, the slopes for the curves at \( x = 1 \) are derived from the tangent line expressions:

• For \( y = x^2 + ax + b \), the slope is \( 2 + a \).
• For \( y = c x - x^2 \), given \( c = 1 \), the slope is \( -1 \).

These slopes must be equal at the common point for the tangent to be shared. Hence, equating \( 2 + a \) to \(-1 \) ensures that the tangent line at \((1,0)\) is the same for both curves. This fundamental understanding helps us solve for the unknown variables by equating their slopes.

Intersection points are where two curves meet or cross each other. In the case of a common tangent, the tangent line "touches" both curves at distinct points, which means it has an equal slope at the point of tangency. For the two curves \( y = x^2 + ax + b \) and \( y = cx - x^2 \), the crucial intersection point is at \((1,0)\). Here, you substitute \( x = 1 \) and \( y = 0 \) into the equations of both curves, producing:

• For \( y = x^2 + ax + b \): \( 1 + a + b = 0 \).
• For \( y = cx - x^2 \): \( c = 1 \).

These substitutions help us determine specific values for the unknown constants in each equation. The intersection concept allows us to validate that the lines do indeed meet at a single, specified point, and thus, share a common tangent at this point.