Question

If V and S are respectively the vertex and focus of the parabola y² + 6y + 2x + 5 = 0, then SV = 

Options

• 2 

• 1/2 

• 1 

• none of these 

Answer 1

1/2 

Given:
The vertex and the focus of a parabola are V and S, respectively.
The given equation of parabola can be rewritten as follows:  

\[\left( y + 3 \right)^2 - 9 + 5 + 2x = 0\] 

\[\Rightarrow \left( y + 3 \right)^2 + 2x = 4\]
\[ \Rightarrow \left( y + 3 \right)^2 = 4 - 2x\]
\[ \Rightarrow \left( y + 3 \right)^2 = - 2\left( x - 2 \right)\] 

Let

\[Y = y + 3, X = x - 2\] 

Then, the equation of parabola becomes \[Y^2 = - 2X\] 

Vertex = \[\left( X = 0, Y = 0 \right) = \left( x - 2 = 0, y + 3 = 0 \right) = \left( x = 2, y = - 3 \right)\] 

Comparing with y² = 4ax:\[4a = 2 \Rightarrow a = \frac{1}{2}\] 

Focus = \[\left( X = \frac{- 1}{2}, Y = 0 \right) = \left( x - 2 = \frac{- 1}{2}, y + 3 = 0 \right) = \left( x = \frac{3}{2}, y = - 3 \right)\]

⇒ SV = \[\sqrt{\left( 2 - \frac{3}{2} \right)^2 + \left( - 3 + 3 \right)^2} = \frac{1}{2}\]