`y^(2) = 4x` is a curve and P, Q and r are three points on it, where P = 1,2, Q = (1/4, 1) and the tangent to the curve at R is parallel to the chord PQ of the curve, then coordinates of R are

Options

A. `((5)/(8), sqrt(5)/(2))`
B. `(9/16, 3/2)`
C. `((5)/(8), -sqrt(5)/(2))`
D. `(9/16, (-3)/(2))`

Answer

To solve the problem step by step, we will follow the reasoning presented in the video transcript:

### Step 1: Understand the curve and points
The given curve is \( y^2 = 4x \). The points \( P \) and \( Q \) are given as \( P(1, 2) \) and \( Q\left(\frac{1}{4}, 1\right) \).

### Step 2: Find the slope of the chord PQ
To find the slope of the chord \( PQ \), we use the formula for the slope between two points:
\[ \text{slope} = \frac{y_2 - y_1}{x_2 - x_1} \]
Substituting the coordinates of points \( P \) and \( Q \):
\[ \text{slope of } PQ = \frac{1 - 2}{\frac{1}{4} - 1} = \frac{-1}{\frac{-3}{4}} = \frac{4}{3} \]

### Step 3: Assume coordinates of point R
Let the coordinates of point \( R \) be \( (h, k) \). Since \( R \) lies on the curve, it must satisfy the equation \( y^2 = 4x \):
\[ k^2 = 4h \]

### Step 4: Find the slope of the tangent at R
The slope of the tangent to the curve at any point can be found by differentiating the equation \( y^2 = 4x \):
\[ \frac{d}{dx}(y^2) = \frac{d}{dx}(4x) \implies 2y \frac{dy}{dx} = 4 \implies \frac{dy}{dx} = \frac{2}{y} \]
At point \( R(h, k) \), the slope of the tangent becomes:
\[ \frac{dy}{dx} = \frac{2}{k} \]

### Step 5: Set the slopes equal
Since the tangent at \( R \) is parallel to the chord \( PQ \), we equate the slopes:
\[ \frac{2}{k} = \frac{4}{3} \]

### Step 6: Solve for k
Cross-multiplying gives:
\[ 2 \cdot 3 = 4k \implies 6 = 4k \implies k = \frac{6}{4} = \frac{3}{2} \]

### Step 7: Substitute k back to find h
Now substitute \( k = \frac{3}{2} \) back into the equation \( k^2 = 4h \):
\[ \left(\frac{3}{2}\right)^2 = 4h \implies \frac{9}{4} = 4h \implies h = \frac{9}{16} \]

### Step 8: Final coordinates of R
Thus, the coordinates of point \( R \) are:
\[ R\left(\frac{9}{16}, \frac{3}{2}\right) \]

### Summary
The coordinates of point \( R \) are \( \left(\frac{9}{16}, \frac{3}{2}\right) \).

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