Question

If $x - 2y + k = 0$ is a tangent to the parabola $y^2 - 4x - 4y + 8 = 0$, then the value of $k$ is:

• A. 2
• B. \(\frac{2}{5}\)
• C. 7
• D. -7

Answer 1

The Correct Option is C

To find the value of \(k\) for which the line \(x - 2y + k = 0\) is tangent to the parabola \(y^2 - 4x - 4y + 8 = 0\), we proceed as follows:

1. Expressing \(x\) from the Line Equation:
The line equation is \(x - 2y + k = 0\). 
\( x = 2y - k \)

2. Substituting into the Parabola Equation:
The parabola equation is \(y^2 - 4x - 4y + 8 = 0\). 
\( y^2 - 4(2y - k) - 4y + 8 = 0 \)
Expand and simplify:

\( y^2 - 8y + 4k - 4y + 8 = 0 \)
\( y^2 - 12y + 4k + 8 = 0 \)

3. Condition for Tangency:
Since the line is tangent to the parabola, there is exactly one point of intersection. This means the quadratic equation in \(y\), \(y^2 - 12y + (4k + 8) = 0\), must have exactly one solution, so its discriminant must be zero.

4. Calculating the Discriminant:
For the quadratic \(y^2 - 12y + (4k + 8) = 0\), the coefficients are \(a = 1\), \(b = -12\), \(c = 4k + 8\). The discriminant is:

\( \Delta = b^2 - 4ac = (-12)^2 - 4 \cdot 1 \cdot (4k + 8) = 144 - 16k - 32 = 112 - 16k \)
Set the discriminant to zero:

\( 112 - 16k = 0 \)

5. Solving for \(k\):
\( 16k = 112 \)
\( k = \frac{112}{16} = 7 \)

Final Answer:
The value of \(k\) for which the line is tangent to the parabola is \(7\).