Number of points of intersectionsof circle `x^(2)+y^(2)+2x=0`
with `y^(2)=4x` is

Options

A. 1
B. 2
C. 3
D. 4

Answer

To find the number of points of intersection between the circle given by the equation \(x^2 + y^2 + 2x = 0\) and the parabola given by the equation \(y^2 = 4x\), we will follow these steps:

### Step 1: Rewrite the Circle's Equation
The equation of the circle can be rewritten by completing the square for the \(x\) terms.

\[ x^2 + 2x + y^2 = 0 \]

Completing the square for \(x^2 + 2x\):

\[ (x + 1)^2 - 1 + y^2 = 0 \]

This simplifies to:

\[ (x + 1)^2 + y^2 = 1 \]

This represents a circle centered at \((-1, 0)\) with a radius of \(1\).

### Step 2: Substitute the Parabola's Equation into the Circle's Equation
Now, we substitute \(y^2 = 4x\) from the parabola into the circle's equation:

\[ (x + 1)^2 + 4x = 1 \]

### Step 3: Simplify the Equation
Expanding and simplifying the equation:

\[ (x + 1)^2 + 4x - 1 = 0 \]

Expanding \((x + 1)^2\):

\[ x^2 + 2x + 1 + 4x - 1 = 0 \]

This simplifies to:

\[ x^2 + 6x = 0 \]

### Step 4: Factor the Quadratic Equation
Factoring the quadratic equation:

\[ x(x + 6) = 0 \]

This gives us two possible values for \(x\):

\[ x = 0 \quad \text{or} \quad x = -6 \]

### Step 5: Find Corresponding \(y\) Values
Now we will find the corresponding \(y\) values for each \(x\).

1. **For \(x = 0\)**:
\[ y^2 = 4(0) = 0 \implies y = 0 \]
Thus, we have the point \((0, 0)\).

2. **For \(x = -6\)**:
\[ y^2 = 4(-6) = -24 \]
Since \(y^2\) cannot be negative, there are no real \(y\) values corresponding to \(x = -6\).

### Conclusion
The only point of intersection is \((0, 0)\). Therefore, the number of points of intersection of the circle and the parabola is:

\[ \text{Number of points of intersection} = 1 \]

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