Let Q be the foot of the perpendicular from the origin O to the tangent at a point `P(alpha, beta)` on the parabola `y^(2)=4ax` and S be the focus of the parabola , then `(OQ)^(2)` (SP) is equal to

Options

A. `alpha`
B. `a alpha^(2)`
C. `beta`
D. `a beta^(2)`

Answer

To solve the problem, we need to find the value of \( (OQ)^2 \cdot (SP) \) where:

- \( O \) is the origin \( (0, 0) \)
- \( P(\alpha, \beta) \) is a point on the parabola \( y^2 = 4ax \)
- \( S(a, 0) \) is the focus of the parabola
- \( Q \) is the foot of the perpendicular from \( O \) to the tangent at point \( P \)

### Step 1: Find the coordinates of point \( P \)
Since \( P \) lies on the parabola \( y^2 = 4ax \), we have:
\[ \beta^2 = 4a\alpha \]

### Step 2: Write the equation of the tangent at point \( P \)
The equation of the tangent to the parabola \( y^2 = 4ax \) at point \( P(\alpha, \beta) \) is given by:
\[ y \beta = 2a(x + \alpha) \]
Rearranging gives:
\[ 2ax - \beta y + 2a\alpha = 0 \]

### Step 3: Find the foot of the perpendicular \( Q \)
The foot of the perpendicular from the origin \( O(0, 0) \) to the line \( 2ax - \beta y + 2a\alpha = 0 \) can be found using the formula for the distance from a point to a line. The distance \( d \) from point \( (x_0, y_0) \) to the line \( Ax + By + C = 0 \) is given by:
\[ d = \frac{|Ax_0 + By_0 + C|}{\sqrt{A^2 + B^2}} \]
Here, \( A = 2a \), \( B = -\beta \), and \( C = 2a\alpha \). Thus, we have:
\[ OQ = \frac{|2a(0) - \beta(0) + 2a\alpha|}{\sqrt{(2a)^2 + (-\beta)^2}} = \frac{|2a\alpha|}{\sqrt{4a^2 + \beta^2}} \]

### Step 4: Substitute \( \beta^2 \)
Using the relation \( \beta^2 = 4a\alpha \), we can substitute \( \beta^2 \) in the denominator:
\[ OQ = \frac{2a\alpha}{\sqrt{4a^2 + 4a\alpha}} = \frac{2a\alpha}{\sqrt{4a(a + \alpha)}} \]
This simplifies to:
\[ OQ = \frac{2a\alpha}{2\sqrt{a(a + \alpha)}} = \frac{a\alpha}{\sqrt{a(a + \alpha)}} \]

### Step 5: Calculate \( (OQ)^2 \)
Now, we calculate \( (OQ)^2 \):
\[ (OQ)^2 = \left(\frac{a\alpha}{\sqrt{a(a + \alpha)}}\right)^2 = \frac{a^2\alpha^2}{a(a + \alpha)} = \frac{a\alpha^2}{a + \alpha} \]

### Step 6: Calculate \( SP \)
The distance \( SP \) from the focus \( S(a, 0) \) to point \( P(\alpha, \beta) \) is given by:
\[ SP = \sqrt{(\alpha - a)^2 + \beta^2} \]
Substituting \( \beta^2 = 4a\alpha \):
\[ SP = \sqrt{(\alpha - a)^2 + 4a\alpha} \]

### Step 7: Calculate \( SP^2 \)
Now, we calculate \( SP^2 \):
\[ SP^2 = (\alpha - a)^2 + 4a\alpha \]
Expanding this gives:
\[ SP^2 = \alpha^2 - 2a\alpha + a^2 + 4a\alpha = \alpha^2 + 2a\alpha + a^2 = (\alpha + a)^2 \]

### Step 8: Final Calculation
Now we need to find \( (OQ)^2 \cdot SP^2 \):
\[ (OQ)^2 \cdot SP^2 = \frac{a\alpha^2}{a + \alpha} \cdot (\alpha + a)^2 = a\alpha^2(\alpha + a) \]

Thus, the final answer is:
\[ \boxed{a\alpha^2(\alpha + a)} \]