if `(x-a)^2+(y-b)^2=c^2` then `({1+((dy)/(dx))^2})^(3/2)/{(d^2y)/(dx^2)} =?`

Options

A. a
B. b
C. -c
D. none of these

Answer

To solve the problem, we start with the equation of a circle given by:

\[ (x - a)^2 + (y - b)^2 = c^2 \]

We need to find the expression:

\[ \frac{(1 + \left(\frac{dy}{dx}\right)^2)^{\frac{3}{2}}}{\frac{d^2y}{dx^2}} \]

### Step 1: Differentiate the equation with respect to \(x\)

Differentiating both sides of the equation with respect to \(x\):

\[ \frac{d}{dx}[(x - a)^2] + \frac{d}{dx}[(y - b)^2] = \frac{d}{dx}[c^2] \]

This gives us:

\[ 2(x - a) + 2(y - b) \frac{dy}{dx} = 0 \]

### Step 2: Solve for \(\frac{dy}{dx}\)

Rearranging the equation, we find:

\[ 2(y - b) \frac{dy}{dx} = -2(x - a) \]

Dividing both sides by \(2(y - b)\):

\[ \frac{dy}{dx} = -\frac{x - a}{y - b} \]

### Step 3: Differentiate \(\frac{dy}{dx}\) to find \(\frac{d^2y}{dx^2}\)

Now we need to differentiate \(\frac{dy}{dx}\) again with respect to \(x\):

Using the quotient rule:

\[ \frac{d^2y}{dx^2} = \frac{(y - b)(-1) - (x - a)\frac{d}{dx}(y - b)}{(y - b)^2} \]

Since \(\frac{d}{dx}(y - b) = \frac{dy}{dx}\), we substitute:

\[ \frac{d^2y}{dx^2} = \frac{-(y - b) - (x - a)\left(-\frac{x - a}{y - b}\right)}{(y - b)^2} \]

This simplifies to:

\[ \frac{d^2y}{dx^2} = \frac{-(y - b) + \frac{(x - a)^2}{y - b}}{(y - b)^2} \]

Combining the terms gives:

\[ \frac{d^2y}{dx^2} = \frac{-(y - b)^2 + (x - a)^2}{(y - b)^3} \]

### Step 4: Substitute into the original expression

Now we substitute \(\frac{dy}{dx}\) into the expression \(1 + \left(\frac{dy}{dx}\right)^2\):

\[ 1 + \left(-\frac{x - a}{y - b}\right)^2 = 1 + \frac{(x - a)^2}{(y - b)^2} \]

This simplifies to:

\[ \frac{(y - b)^2 + (x - a)^2}{(y - b)^2} = \frac{c^2}{(y - b)^2} \]

### Step 5: Calculate the final expression

Now we can substitute back into the original expression:

\[ \frac{(1 + \left(\frac{dy}{dx}\right)^2)^{\frac{3}{2}}}{\frac{d^2y}{dx^2}} = \frac{\left(\frac{c^2}{(y - b)^2}\right)^{\frac{3}{2}}}{\frac{-(y - b)^2 + (x - a)^2}{(y - b)^3}} \]

This simplifies to:

\[ \frac{\frac{c^3}{(y - b)^3}}{\frac{-(y - b)^2 + (x - a)^2}{(y - b)^3}} = \frac{c^3}{-(y - b)^2 + (x - a)^2} \]

Since \(-(y - b)^2 + (x - a)^2 = -c^2\) (from the original equation), we find:

\[ \frac{c^3}{-c^2} = -c \]

### Final Answer

Thus, the value of the expression is:

\[ -c \]