If a,b,c form a GP with common ratio r, the sum of the ordinates of the points of intersection of the line ax+by+c=0 and the curve `x+2y^2=0` is

Options

A. a) `(-r^(2))/(2)`
B. b) `-(r)/(2)`
C. c)`(r)/(2)`
D. d) r

Answer

To solve the problem, we need to find the sum of the ordinates of the points of intersection of the line \( ax + by + c = 0 \) and the curve \( x + 2y^2 = 0 \), given that \( a, b, c \) are in geometric progression (GP) with common ratio \( r \).

### Step-by-Step Solution:

1. **Identify the equations**:
The line is given by:
\[ ax + by + c = 0 \quad \text{(Equation 1)} \]
The curve is given by:
\[ x + 2y^2 = 0 \quad \text{(Equation 2)} \]

2. **Express \( b \) and \( c \) in terms of \( a \)**:
Since \( a, b, c \) are in GP with common ratio \( r \), we can express:
\[ b = ar \quad \text{and} \quad c = ar^2 \]

3. **Substitute \( b \) and \( c \) into the line equation**:
Substituting these values into Equation 1 gives:
\[ ax + ar y + ar^2 = 0 \]
Factoring out \( a \) (assuming \( a \neq 0 \)):
\[ a(x + ry + r^2) = 0 \implies x + ry + r^2 = 0 \]
Thus, we have:
\[ x = -ry - r^2 \quad \text{(Equation 3)} \]

4. **Substitute \( x \) from Equation 3 into Equation 2**:
Substitute \( x \) into the curve equation:
\[ -ry - r^2 + 2y^2 = 0 \]
Rearranging gives:
\[ 2y^2 - ry - r^2 = 0 \]

5. **Solve the quadratic equation for \( y \)**:
Using the quadratic formula \( y = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} \):
Here, \( A = 2 \), \( B = -r \), and \( C = -r^2 \):
\[ y = \frac{-(-r) \pm \sqrt{(-r)^2 - 4 \cdot 2 \cdot (-r^2)}}{2 \cdot 2} \]
Simplifying this:
\[ y = \frac{r \pm \sqrt{r^2 + 8r^2}}{4} = \frac{r \pm \sqrt{9r^2}}{4} = \frac{r \pm 3r}{4} \]

6. **Calculate the two values of \( y \)**:
The two solutions for \( y \) are:
\[ y_1 = \frac{4r}{4} = r \quad \text{and} \quad y_2 = \frac{-2r}{4} = -\frac{r}{2} \]

7. **Find the sum of the ordinates**:
The sum of the ordinates (the \( y \)-values) is:
\[ y_1 + y_2 = r - \frac{r}{2} = \frac{2r}{2} - \frac{r}{2} = \frac{r}{2} \]

### Final Answer:
The sum of the ordinates of the points of intersection is:
\[ \frac{r}{2} \]