The ends of a line segment are `P(1, 3) and Q(1,1)`, R is a point on the line segment PQ such that `PR : QR=1:lambda`.If R is an interior point of the parabola `y^2=4x` then

Options

A. (0, 1)
B. (-3/5, 1)
C. (1/2, 3/5)
D. none of these

Answer

To solve the problem step by step, we will first find the coordinates of point R on the line segment PQ and then determine the conditions under which R is an interior point of the parabola \(y^2 = 4x\).

### Step 1: Determine the coordinates of point R
Given the points \(P(1, 3)\) and \(Q(1, 1)\), we need to find the coordinates of point R such that the ratio \(PR : QR = 1 : \lambda\).

Let the coordinates of point R be \((1, y)\) since both P and Q have the same x-coordinate.

Using the section formula, the coordinates of R can be expressed as:
\[ R = \left(\frac{m x_2 + n x_1}{m+n}, \frac{m y_2 + n y_1}{m+n}\right) \]
where \(m = 1\), \(n = \lambda\), \(P(1, 3)\) is \((x_1, y_1)\), and \(Q(1, 1)\) is \((x_2, y_2)\).

Substituting the values:
\[ R = \left(1, \frac{1 \cdot 1 + \lambda \cdot 3}{1 + \lambda}\right) = \left(1, \frac{3\lambda + 1}{\lambda + 1}\right) \]

### Step 2: Determine the condition for R to be an interior point of the parabola
For R to be an interior point of the parabola \(y^2 = 4x\), the point must satisfy the inequality:
\[ y^2 < 4x \]

Substituting the coordinates of R into the inequality:
\[ \left(\frac{3\lambda + 1}{\lambda + 1}\right)^2 < 4 \cdot 1 \]
This simplifies to:
\[ \left(\frac{3\lambda + 1}{\lambda + 1}\right)^2 < 4 \]

### Step 3: Solve the inequality
Taking the square root of both sides (considering only the positive root since y is positive):
\[ \frac{3\lambda + 1}{\lambda + 1} < 2 \]
Cross-multiplying gives:
\[ 3\lambda + 1 < 2(\lambda + 1) \]
Expanding the right side:
\[ 3\lambda + 1 < 2\lambda + 2 \]
Rearranging gives:
\[ 3\lambda - 2\lambda < 2 - 1 \]
\[ \lambda < 1 \]

### Step 4: Find the lower bound for λ
Now, we also need to ensure that the point R is between P and Q. Since R divides PQ in the ratio \(1 : \lambda\), \(\lambda\) must be greater than 0.

### Conclusion
Thus, the value of \(\lambda\) must satisfy:
\[ 0 < \lambda < 1 \]

### Final Answer
The final answer is:
\[ \lambda \in (0, 1) \]