Question

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 \[x + 2{y^2} = 0\] is
A. \[ - \dfrac{{{r^2}}}{2}\]
B. \[ - \dfrac{r}{2}\]
C. \[\dfrac{r}{2}\]
D. \[\dfrac{{{r^2}}}{2}\]

Answer 1

Hint: Geometric Progression (GP) is a type of sequence where each succeeding term is produced by multiplying each preceding term by a fixed number, which is called a common ratio. This progression is also known as a geometric sequence of numbers that follow a pattern.

Complete step-by-step answer:
General form of a GP
 \[a,ar,a{r^2},a{r^3},...,a{r^n}\]
where \[a\] is the first term
 \[r\] is the common ratio
 \[{r^n}\] is the last term
If the common ratio is:
Negative: the result will alternate between positive and negative.
Greater than \[1\] : there will be an exponential growth towards infinity (positive).
Less than \[ - 1\] : there will be an exponential growth towards infinity (positive and negative).
Between \[1\] and \[ - 1\] : there will be an exponential decay towards zero.
Zero: the result will remain at zero
When three quantities are in GP, the middle one is called the geometric mean of the other two.
Let \[b = ar\] and \[c = a{r^2}\]
Putting the values of \[b\] and \[c\] in the given equation of line we get ,
 \[ax + ary + a{r^2} = 0\]
Simplifying this we get ,
 \[x + ry + {r^2} = 0\]
Hence we get ,
 \[x = - ry - {r^2}\]
Now putting the value of \[x\] in the given equation of curve we get ,
 \[ - ry - {r^2} + 2{y^2} = 0\]
Which can be rewritten as \[2{y^2} - ry - {r^2} = 0\]
This is a quadratic equation in \[y\]
Therefore using the quadratic formula we get ,
 \[y = \dfrac{{ - b \pm \sqrt {{b^2} - 4ac} }}{{2a}}\]
 \[y = \dfrac{{ - ( - r) \pm \sqrt {{{( - r)}^2} - 4(2)( - {r^2})} }}{{2(2)}}\]
Which on simplification gives us ,
 \[y = \dfrac{{r \pm \sqrt {{r^2} + 8{r^2}} }}{4}\]
Hence we get \[y = r\] or \[y = - \dfrac{r}{2}\]
Therefore sum of ordinates \[ = r + \left( { - \dfrac{r}{2}} \right)\]
Hence sum of ordinates \[ = \dfrac{r}{2}\]
So, the correct answer is “Option C”.

Note: Geometric Progression (GP) is a type of sequence where each succeeding term is produced by multiplying each preceding term by a fixed number, which is called a common ratio. Remember the quadratic formula to find roots of an equation.