The equation of tangent at (8,8) to the parabola `y^2=8x` is ?

Answer

To find the equation of the tangent to the parabola \( y^2 = 8x \) at the point \( (8, 8) \), we can follow these steps:

### Step 1: Identify the parabola and the point
The given parabola is \( y^2 = 8x \) and the point at which we need to find the tangent is \( (8, 8) \).

### Step 2: Use the formula for the equation of the tangent
For a parabola of the form \( y^2 = 4ax \), the equation of the tangent at the point \( (x_1, y_1) \) is given by:
\[ yy_1 = 2a(x + x_1) \]
Here, \( 4a = 8 \) implies \( a = 2 \).

### Step 3: Substitute the values into the tangent formula
Now, substituting \( a = 2 \), \( x_1 = 8 \), and \( y_1 = 8 \) into the tangent equation:
\[ y \cdot 8 = 2 \cdot 2 (x + 8) \]
This simplifies to:
\[ 8y = 8(x + 8) \]

### Step 4: Simplify the equation
Dividing both sides by 8 gives:
\[ y = x + 8 \]

### Step 5: Rearranging the equation
Rearranging the equation gives us:
\[ x - y + 8 = 0 \]

### Final Answer
Thus, the equation of the tangent at the point \( (8, 8) \) to the parabola \( y^2 = 8x \) is:
\[ x - y + 8 = 0 \]

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