Tangents are drawn to the hyperbola `x^2/9-y^2/4=1` parallet to the sraight line `2x-y=1.` The points of contact of the tangents on the hyperbola are (A) `(9/(2sqrt2),1/sqrt2)` (B) `(-9/(2sqrt2),-1/sqrt2)` (C) `(3sqrt3,-2sqrt2)` (D) `(-3sqrt3,2sqrt2)`

Options

A. `((9)/(2sqrt2),(1)/(sqrt2))`
B. `(-(9)/(2sqrt2),-(1)/(sqrt2))`
C. `(3sqrt3,-2sqrt2)`
D. `(3sqrt3,-2sqrt2)`

Explanation

To solve the problem of finding the points of contact of tangents drawn to the hyperbola \( \frac{x^2}{9} - \frac{y^2}{4} = 1 \) that are parallel to the line \( 2x - y = 1 \), we can follow these steps:

### Step 1: Identify the slope of the given line
The equation of the line is \( 2x - y = 1 \). We can rewrite this in slope-intercept form \( y = mx + c \):
\[ y = 2x - 1 \]
From this, we can see that the slope \( m \) of the line is \( 2 \).
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