The point on the parabola `y^(2)=4x` at which it cuts the straight line joining (0,0) and (2,3) is

Options

A. (0,0)
B. (2,3)
C. (16/9,8/3)
D. (1,2)

Answer

To find the point on the parabola \( y^2 = 4x \) at which it cuts the straight line joining the points \( (0,0) \) and \( (2,3) \), we can follow these steps:

### Step 1: Find the equation of the line
The line passes through the points \( (0,0) \) and \( (2,3) \). We can use the two-point form of the equation of a line:

\[ y - y_1 = \frac{y_2 - y_1}{x_2 - x_1} (x - x_1) \]

Substituting \( (x_1, y_1) = (0, 0) \) and \( (x_2, y_2) = (2, 3) \):

\[ y - 0 = \frac{3 - 0}{2 - 0} (x - 0) \]

This simplifies to:

\[ y = \frac{3}{2} x \]

### Step 2: Substitute the line equation into the parabola equation
Now, we have the parabola \( y^2 = 4x \) and the line \( y = \frac{3}{2} x \). We will substitute the expression for \( y \) from the line into the parabola's equation:

\[ \left(\frac{3}{2} x\right)^2 = 4x \]

### Step 3: Simplify the equation
Expanding the left side:

\[ \frac{9}{4} x^2 = 4x \]

To eliminate the fraction, multiply through by 4:

\[ 9x^2 = 16x \]

### Step 4: Rearrange the equation
Rearranging gives us a standard quadratic equation:

\[ 9x^2 - 16x = 0 \]

### Step 5: Factor the equation
Factoring out \( x \):

\[ x(9x - 16) = 0 \]

This gives us two solutions:

1. \( x = 0 \)
2. \( 9x - 16 = 0 \) which gives \( x = \frac{16}{9} \)

### Step 6: Find corresponding \( y \) values
Now, we will find the corresponding \( y \) values for both \( x \) values using the line equation \( y = \frac{3}{2} x \):

1. For \( x = 0 \):
\[ y = \frac{3}{2} \cdot 0 = 0 \quad \Rightarrow \quad (0, 0) \]

2. For \( x = \frac{16}{9} \):
\[ y = \frac{3}{2} \cdot \frac{16}{9} = \frac{48}{18} = \frac{8}{3} \quad \Rightarrow \quad \left(\frac{16}{9}, \frac{8}{3}\right) \]

### Step 7: Conclusion
Thus, the points where the line intersects the parabola are:

1. \( (0, 0) \)
2. \( \left(\frac{16}{9}, \frac{8}{3}\right) \)

### Final Answer
The points on the parabola \( y^2 = 4x \) at which it cuts the straight line joining \( (0,0) \) and \( (2,3) \) are \( (0, 0) \) and \( \left(\frac{16}{9}, \frac{8}{3}\right) \).

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