Question

The equation of the tangent to the parabola $y^2 = 8x$ which is parallel to the line $2x - y + 5 = 0$ is:

• A. $2x - y + 1 = 0$
• B. $2x - y - 1 = 0$
• C. $2x - y + 2 = 0$
• D. $2x - y - 2 = 0$

Hint:

The condition for a line $y = mx + c$ to be tangent to the parabola $y^2 = 4ax$ is $c = \frac{a}{m}$.

Answer 1

The Correct Option is A

Step 1: Recall the slope tangent form.
For the parabola $y^2 = 4ax$, a tangent with slope $m$ has the equation $y = mx + \frac{a}{m}$. So we need both $a$ and $m$.

Step 2: Find $a$ from the parabola.
The parabola is $y^2 = 8x$. Compare with $y^2 = 4ax$ to get $4a = 8$, so $a = 2$.

Step 3: Find the slope $m$.
The tangent must be parallel to $2x - y + 5 = 0$. Rearranged, this is $y = 2x + 5$, so its slope is $2$. Parallel means the same slope, so $m = 2$.

Step 4: Put values into the tangent form.
Substitute $a = 2$ and $m = 2$.
\[ y = 2x + \frac{2}{2} = 2x + 1 \]

Step 5: Rearrange to standard form.
Move all terms to one side.
\[ 2x - y + 1 = 0 \]

Step 6: State the answer.
So the required tangent is $2x - y + 1 = 0$.
\[ \boxed{2x - y + 1 = 0} \]