Let tangents at P and Q to curve `y^(2)-4x-2y+5=0` intersect at T. If S(2, 1) is a point such that `(SP)(SQ)=16`, then the length ST is equal to :

Options

A. 3
B. 4
C. 5
D. None of these

Answer

To solve the problem step by step, we will first rewrite the equation of the parabola and then use the properties of tangents to find the required length.

### Step 1: Rewrite the equation of the parabola
The given equation of the parabola is:
\[ y^2 - 4x - 2y + 5 = 0 \]

We can rearrange this to isolate \(y\):
\[ y^2 - 2y = 4x - 5 \]

Now, we complete the square for the left-hand side:
\[ (y - 1)^2 - 1 = 4x - 5 \]
\[ (y - 1)^2 = 4x - 4 \]
\[ (y - 1)^2 = 4(x - 1) \]

This shows that the parabola opens to the right with vertex at \((1, 1)\).

### Step 2: Identify the focus of the parabola
From the standard form \((y - k)^2 = 4a(x - h)\), we can see that:
- Vertex \((h, k) = (1, 1)\)
- The value of \(4a = 4\) implies \(a = 1\).

The focus of the parabola is located at:
\[ (h + a, k) = (1 + 1, 1) = (2, 1) \]

### Step 3: Use the property of tangents
Let \(S(2, 1)\) be a point from which tangents are drawn to the parabola at points \(P\) and \(Q\). According to the property of tangents from a point to a parabola, we have:
\[ SP \cdot SQ = ST^2 \]
where \(T\) is the point of intersection of the tangents at \(P\) and \(Q\).

### Step 4: Substitute the given information
We are given that:
\[ SP \cdot SQ = 16 \]

Using the property:
\[ ST^2 = 16 \]

### Step 5: Solve for \(ST\)
Taking the square root of both sides, we find:
\[ ST = \sqrt{16} = 4 \]

Thus, the length \(ST\) is equal to \(4\).

### Final Answer
The length \(ST\) is \(4\).

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