find the constants a,b and c so that the two curves y=x² + ax + b and y= cx – x² will touch each other at the pont (1,0)

ans is : a= -3 , b=2, c=1

To find the constants \( a \), \( b \), and \( c \) such that the curves \( y = x^2 + ax + b \) and \( y = cx - x^2 \) touch each other at the point \( (1, 0) \), we need to ensure two conditions are met: the curves must intersect at that point, and they must have the same slope at that point. Let's break this down step by step.

Step 1: Setting Up the Equations

We have two equations:

• Curve 1: \( y = x^2 + ax + b \)
• Curve 2: \( y = cx - x^2 \)

Step 2: Ensuring the Curves Intersect at (1, 0)

For the curves to touch at the point \( (1, 0) \), both equations must equal zero when \( x = 1 \).

Substituting \( x = 1 \) into Curve 1:

\( 0 = 1^2 + a(1) + b \)

This simplifies to:

\( 0 = 1 + a + b \)

So, we have our first equation:

Equation 1: \( a + b = -1 \)

Now, substituting \( x = 1 \) into Curve 2:

\( 0 = c(1) - (1^2) \)

This simplifies to:

\( 0 = c - 1 \)

Thus, we find:

Equation 2: \( c = 1 \)

Step 3: Finding the Slopes

Next, we need to ensure that the slopes of the two curves are equal at \( x = 1 \). We find the derivatives of both equations.

For Curve 1:

\( \frac{dy}{dx} = 2x + a \)

Substituting \( x = 1 \):

\( \frac{dy}{dx} = 2(1) + a = 2 + a \)

For Curve 2:

\( \frac{dy}{dx} = c - 2x \)

Substituting \( x = 1 \):

\( \frac{dy}{dx} = c - 2(1) = c - 2 \)

Step 4: Setting the Slopes Equal

Since the slopes must be equal at the point of tangency, we set the derivatives equal to each other:

From Curve 1: \( 2 + a \)

From Curve 2: \( c - 2 \)

Setting them equal gives us:

Equation 3: \( 2 + a = c - 2 \)

Step 5: Substituting Known Values

We already know \( c = 1 \). Substituting this into Equation 3:

\( 2 + a = 1 - 2 \)

This simplifies to:

\( 2 + a = -1 \)

Solving for \( a \):

\( a = -1 - 2 = -3 \)

Step 6: Finding b

Now that we have \( a = -3 \), we can substitute this back into Equation 1 to find \( b \):

\( -3 + b = -1 \)

Thus:

\( b = -1 + 3 = 2 \)

Final Values

We have determined the constants:

• a: -3
• b: 2
• c: 1

In summary, the values of \( a \), \( b \), and \( c \) that ensure the two curves touch at the point \( (1, 0) \) are \( a = -3 \), \( b = 2 \), and \( c = 1 \). This process illustrates how to analyze the conditions for tangency between two curves through intersection points and matching slopes.