Let `A` be a point on the `x` -axis.Common tangents are drawn from `A` to the curves `x^(2)+y^(2)=8` and `y^(2)=16x`.If one of these tangents touches the two curves at `Q` and `R`,then `(QR)^(2)` is equal to

Options

A. `76`
B. `81`
C. `72`
D. `64`

Answer

To solve the problem, we need to find the square of the distance \( (QR)^2 \) where \( Q \) and \( R \) are the points of tangency from point \( A \) on the \( x \)-axis to the curves given by the equations \( x^2 + y^2 = 8 \) (a circle) and \( y^2 = 16x \) (a parabola).

### Step 1: Identify the curves
The first curve is a circle centered at the origin with radius \( \sqrt{8} \) and the second curve is a parabola that opens to the right.

### Step 2: Determine the point \( A \)
Let \( A \) be the point \( (a, 0) \) on the \( x \)-axis.

### Step 3: Find the equation of the tangent to the circle
The equation of the tangent to the circle \( x^2 + y^2 = 8 \) from point \( A(a, 0) \) can be expressed as:
\[ y = mx + c \]
where \( c = \sqrt{8 - a^2} \) (using the formula for the tangent line).

### Step 4: Find the equation of the tangent to the parabola
For the parabola \( y^2 = 16x \), the equation of the tangent line can be written as:
\[ y = mx + \frac{4}{m} \]
where \( m \) is the slope of the tangent.

### Step 5: Set up the condition for common tangents
For the tangents to be common, the tangents must satisfy both curves. Therefore, we equate the two expressions for \( y \):
\[ mx + \sqrt{8 - a^2} = mx + \frac{4}{m} \]
This leads to:
\[ \sqrt{8 - a^2} = \frac{4}{m} \]

### Step 6: Substitute and solve for \( m \)
From the circle's tangent condition, we also have:
\[ \frac{4}{m^2} = 8(1 + m^2) \]
This simplifies to:
\[ 4 = 8m^2 + 8m^4 \]
Rearranging gives:
\[ 8m^4 + 8m^2 - 4 = 0 \]
Dividing through by 4:
\[ 2m^4 + 2m^2 - 1 = 0 \]
Let \( x = m^2 \):
\[ 2x^2 + 2x - 1 = 0 \]
Using the quadratic formula:
\[ x = \frac{-2 \pm \sqrt{(2)^2 - 4 \cdot 2 \cdot (-1)}}{2 \cdot 2} = \frac{-2 \pm \sqrt{4 + 8}}{4} = \frac{-2 \pm \sqrt{12}}{4} = \frac{-2 \pm 2\sqrt{3}}{4} = \frac{-1 \pm \sqrt{3}}{2} \]
Thus, \( m^2 = \frac{-1 + \sqrt{3}}{2} \) or \( m^2 = \frac{-1 - \sqrt{3}}{2} \) (only the positive root is valid).

### Step 7: Calculate points \( Q \) and \( R \)
Using \( m \) values, we can find the points of tangency \( Q \) and \( R \) on the circle and parabola respectively.

### Step 8: Calculate the distance \( QR \)
The distance \( QR \) can be calculated using the distance formula:
\[ QR = \sqrt{(x_Q - x_R)^2 + (y_Q - y_R)^2} \]
Finally, we need to square this distance to find \( (QR)^2 \).

### Step 9: Final calculation
After substituting the coordinates of \( Q \) and \( R \) back into the distance formula, we will find the value of \( (QR)^2 \).

### Conclusion
The final answer for \( (QR)^2 \) is \( 72 \).