Question

If $a,b,c$ form a Geometric Progression(G.P.) 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

• A. $\frac{-r^2}{2}$
• B. $\frac{-r}{2}$
• C. $\frac{r}{2}$
• D. $\frac{r^2}{2}$

Answer 1

The correct option is C

$\frac{r}{2}$

Explanation for the correct option:

Step 1: Calculate the value of $x$

Given $a,b,c$ form a Geometric Progression(G.P.) with a common ratio $r$. So $b=ar$, $c=a{r}^2$

Equation of line is $ax+by+c=0....(1)$

Put $b$ and $c$ in $(1)$

$\begin{array}{ll}\Rightarrow ax+ary+a{r}^2& =0\\ \Rightarrow x& =-r^2–ry\end{array}$

Step 2: Substitute the value of $x$ in curve equation $x+2y^2=0$

$\begin{array}{llrl}⟹-r^2–ry+2y^2& =0& & \\ ⟹2y^2–ry–r^2& =0& & \\ ⟹y& =r\pm \sqrt{\frac{r^2+8r^2}{4}}& & \\ & =\frac{4r}{4}\text{or}\frac{-2r}{4}& & \\ & =r,\frac{-r}{2}& & \end{array}$

Step 3: Find the sum of the ordinates of the points of intersection of the line $ax+by+c=0$ and the curve $x+2y^2=0$

$\begin{array}{rl}\text{Sum of ordinates}& =r–\frac{r}{2}\\ & =\frac{r}{2}\end{array}$

Hence sum of ordinates is $\frac{r}{2}$. Hence option (C) is the correct option.