The point of contact 8x-9y+5 = 0 with the ellipse `4x^(2)+9y^(2)=1` is

Options

A. `(2/5,1/5)`
B. `(-2/5,-1/5)`
C. `(-2/5,1/5)`
D. `(2/5,1/5)`

Answer

To find the point of contact of the tangent line \(8x - 9y + 5 = 0\) with the ellipse \(4x^2 + 9y^2 = 1\), we can follow these steps:

### Step 1: Identify the equations
We have the equation of the ellipse:
\[ 4x^2 + 9y^2 = 1 \]
And the equation of the tangent line:
\[ 8x - 9y + 5 = 0 \]

### Step 2: Write the general form of the tangent to the ellipse
For the ellipse \(ax^2 + by^2 = 1\), the equation of the tangent at the point \((h, k)\) is given by:
\[ \frac{4h}{1}x + \frac{9k}{1}y = 1 \]
This simplifies to:
\[ 4hx + 9ky = 1 \]

### Step 3: Compare the tangent equation with the given line
We can rewrite the tangent equation as:
\[ 4hx + 9ky - 1 = 0 \]
Now, we compare this with the given tangent line equation:
\[ 8x - 9y + 5 = 0 \]

### Step 4: Set up equations from the comparison
From the comparison of coefficients, we have:
1. \(4h = 8\)
2. \(9k = -9\)
3. \(-1 = 5\) (which is not used directly)

From the first two equations, we can solve for \(h\) and \(k\).

### Step 5: Solve for \(h\) and \(k\)
From \(4h = 8\):
\[ h = \frac{8}{4} = 2 \]

From \(9k = -9\):
\[ k = \frac{-9}{9} = -1 \]

### Step 6: Find the point of contact
Thus, the point of contact \((h, k)\) is:
\[ (h, k) = (2, -1) \]

### Final Answer
The point of contact is:
\[ (2, -1) \]