The line ax + by = 1 cute ellipse `cx^(2) + dy^(2) = 1` only once if

Options

A. `ca^(2)+db^(2)=1`
B. `c/a^(2)+d/b^(2)=1`
C. `a^(2)/c+b^(2)/d=1`
D. `ac^(2)+bd^(2)=1`

Answer

To determine the condition under which the line \( ax + by = 1 \) intersects the ellipse \( cx^2 + dy^2 = 1 \) only once, we can follow these steps:

### Step 1: Rewrite the line equation
We start with the line equation:
\[ ax + by = 1 \]
We can express \( y \) in terms of \( x \):
\[ y = \frac{1 - ax}{b} \]

### Step 2: Substitute \( y \) into the ellipse equation
Next, we substitute \( y \) into the ellipse equation:
\[ cx^2 + dy^2 = 1 \]
Substituting for \( y \):
\[ cx^2 + d\left(\frac{1 - ax}{b}\right)^2 = 1 \]

### Step 3: Expand the equation
Now we expand the equation:
\[ cx^2 + d\left(\frac{(1 - ax)^2}{b^2}\right) = 1 \]
This becomes:
\[ cx^2 + \frac{d}{b^2}(1 - 2ax + a^2x^2) = 1 \]
Combining the terms gives:
\[ cx^2 + \frac{d}{b^2} - \frac{2dax}{b^2} + \frac{da^2x^2}{b^2} = 1 \]

### Step 4: Rearranging the equation
Rearranging the equation, we get:
\[ \left(c + \frac{da^2}{b^2}\right)x^2 - \frac{2dax}{b^2} + \left(\frac{d}{b^2} - 1\right) = 0 \]

### Step 5: Identify coefficients
Let:
- \( P = c + \frac{da^2}{b^2} \)
- \( Q = -\frac{2da}{b^2} \)
- \( R = \frac{d}{b^2} - 1 \)

The equation can now be written as:
\[ Px^2 + Qx + R = 0 \]

### Step 6: Apply the condition for a single intersection
For the line to intersect the ellipse only once, the discriminant of this quadratic equation must be zero:
\[ Q^2 - 4PR = 0 \]
Substituting the values of \( P \), \( Q \), and \( R \):
\[ \left(-\frac{2da}{b^2}\right)^2 - 4\left(c + \frac{da^2}{b^2}\right)\left(\frac{d}{b^2} - 1\right) = 0 \]

### Step 7: Simplifying the discriminant
Calculating the discriminant:
\[ \frac{4d^2a^2}{b^4} - 4\left(c + \frac{da^2}{b^2}\right)\left(\frac{d - b^2}{b^2}\right) = 0 \]
This simplifies to:
\[ 4d^2a^2 - 4b^4\left(c + \frac{da^2}{b^2}\right)(d - b^2) = 0 \]

### Step 8: Final condition
From the above, we derive:
\[ a^2d + b^2c = cd \]
Dividing through by \( cd \) gives:
\[ \frac{a^2}{c} + \frac{b^2}{d} = 1 \]

### Conclusion
Thus, the condition for the line \( ax + by = 1 \) to intersect the ellipse \( cx^2 + dy^2 = 1 \) only once is:
\[ \frac{a^2}{c} + \frac{b^2}{d} = 1 \]