If three distinct numbers a,b,c are in G.P. and the equations ax^2 + 2bx + c = 0 and dx^2 + 2ex + f

Q: Why does the first equation only have one root?

Great question! When we calculate the discriminant $D = (2b)^2 - 4ac$, and substitute the G.P. condition $b^2 = ac$, we get $D = 4ac - 4ac = 0$. A quadratic equation with a discriminant of zero is a perfect square, meaning it has two real and equal roots. Geometrically, the parabola just touches the x-axis at one point, so there is only one unique value for the root.

Q: Can we use the common root condition for two quadratics instead of this?

You absolutely could! The condition for two quadratics $a_1x^2 + b_1x + c_1 = 0$ and $a_2x^2 + b_2x + c_2 = 0$ to have a common root is quite complex. However, because the first equation is a perfect square, the 'common root' is simply the root of the first equation. This makes the substitution method much faster and more elegant than the general formula.

Q: How do I know when to divide by $c$ or $ac$ during the simplification?

Always look at the target! The options provided in the question involve terms like $d/a, e/b,$ and $f/c$. This is your roadmap. When you have the equation $d(b^2/a^2) - 2e(b/a) + f = 0$, you can see that dividing by $c$ (and using $b^2=ac$) will naturally produce the denominators $a, b,$ and $c$ that you need to match the options.

Question

If three distinct numbers a,b,c are in G.P. and the equations $ax^2 + 2bx + c = 0$ and $dx^2 + 2ex + f = 0$ have a common root, then which one of the following statements is correct?

Options:

(A) $d,e,f$ are in A.P.

(B) $d/a, e/b, f/c$ are in G.P.

(C) $d/a, e/b, f/c$ are in A.P.

(D) $d,e,f$ are in G.P.

Sigma Percentile · Accepted Answer

### Analyzing the Setup

Imagine you are standing at the intersection of two beautiful mathematical landscapes: the rhythmic, multiplicative world of Geometric Progressions and the geometric, parabolic world of quadratic equations. We are given three distinct numbers $a, b, c$ in a Geometric Progression, and two quadratic equations: $ax^2 + 2bx + c = 0$ and $dx^2 + 2ex + f = 0$.

We are told they share a common root. Our mission is to find the relationship between $d, e,$ and $f$.

### The G.P. Master Key

Let us begin by unpacking the first piece of information. We are told that $a, b, c$ are in a Geometric Progression.

By definition, the middle term squared must equal the product of the other two:

$$b^2 = ac$$

Keep this locked in your memory; it is the secret ingredient that will simplify everything later.

### The Discriminant's Secret

Now, look at the first quadratic equation: $ax^2 + 2bx + c = 0$. To understand its roots, we calculate the discriminant $D = (2b)^2 - 4(a)(c)$.

Expanding this, we get $D = 4b^2 - 4ac$. Now, substitute our master key $b^2 = ac$ into this expression:

$$D = 4(ac) - 4ac = 0$$

This is a moment of pure mathematical elegance! A discriminant of zero means our parabola doesn't just cross the x-axis; it kisses it. It has one unique, repeated root.

Using the quadratic formula, this root is $x = -\frac{2b}{2a}$, which simplifies beautifully to:

$$x = -\frac{b}{a}$$

### The Common Root Bridge

The problem states that our two equations share a common root. Since the first equation only has one unique root, $x = -\frac{b}{a}$, this MUST be the root that the second equation also possesses.

This means that if we plug $x = -\frac{b}{a}$ into the second equation $dx^2 + 2ex + f = 0$, it must satisfy the equation perfectly:

$$d\left(-\frac{b}{a}\right)^2 + 2e\left(-\frac{b}{a}\right) + f = 0$$

Simplifying this, we get:

$$d\left(\frac{b^2}{a^2}\right) - 2e\left(\frac{b}{a}\right) + f = 0$$

### The Algebraic Dance

We are almost there. Let us bring back our master key $b^2 = ac$ and substitute it into the first term:

$$d\left(\frac{ac}{a^2}\right) - 2e\left(\frac{b}{a}\right) + f = 0$$

One $a$ cancels out, leaving:

$$d\left(\frac{c}{a}\right) - 2e\left(\frac{b}{a}\right) + f = 0$$

To reach the desired form, we divide the entire equation by $c$:

$$\frac{d}{a} - 2e\left(\frac{b}{ac}\right) + \frac{f}{c} = 0$$

Again, substitute $ac = b^2$ in the denominator of the middle term:

$$\frac{d}{a} - 2e\left(\frac{b}{b^2}\right) + \frac{f}{c} = 0$$

This simplifies to:

$$\frac{d}{a} - \frac{2e}{b} + \frac{f}{c} = 0$$

### Final Conclusion

Rearranging this, we get:

$$\frac{d}{a} + \frac{f}{c} = \frac{2e}{b}$$

This is the classic condition for an Arithmetic Progression! If three numbers $x, y, z$ are in A.P., then $x + z = 2y$.

Here, our terms are $\frac{d}{a}, \frac{e}{b}, \frac{f}{c}$. Thus, **they are in an A.P.** We have successfully bridged the gap between G.P. and A.P. using the elegance of quadratic roots.

Animated Solution for Mathematics - Sequence and Series: If three distinct numbers a,b,c are in G.P. and the equations $a x^{2} + 2 b x + c = 0$ and $d x^{2} + 2 e x + f = 0$ have a common root, then which one of the following statements is correct?

Select Answer:

Visualized Solution

The G.P. Condition

Analyzing the First Equation

Finding the Discriminant Value

Equal Roots Property

Calculating the Root

The Common Root Concept

Substitution into Second Equation

Simplifying the Terms

Applying the G.P. Condition Again

Atomic Simplification

Dividing by c

Final Substitution

The A.P. Relationship

Conclusion

The Sigma Insight: Geometric Progression (G.P.)

Analyzing the Setup

The G.P

The Discriminant's Secret

The Common Root Bridge

The Algebraic Dance

Final Conclusion

Similar Questions

Let $a, b, c, d$ and $p$ be any non zero distinct real numbers such that $(a^{2} + b^{2} + c^{2}) p^{2} -2 (ab + b c + c d) p + (b^{2} + c^{2} + d^{2}) = 0$ . Then :

If a, b, c are in A.P. and $a^{2}, b^{2}, c^{2}$ are in G.P. such that $a < b < c$ and $a + b + c = \frac{3}{4}$ , then the value of a is :-

If $a, b, c, d$ and $p$ are distinct real numbers such that $(a^{2} + b^{2} + c^{2}) p^{2} - 2 (ab + b c + c d) p + (b^{2} + c^{2} + d^{2}) \leq 0$ then $a, b, c, d$

If $a_{1} (> 0), a_{2}, a_{3}, a_{4}, a_{5}$ are in a G.P., $a_{2} + a_{4} = 2 a_{3} + 1$ and $3 a_{2} + a_{3} = 2 a_{4}$ , then $a_{2} + a_{4} + 2 a_{5}$ is equal to ______.

Let $a$ and $b$ be two distinct positive real numbers. Let $1 1^{t h}$ term of a GP, whose first term is $a$ and third term is $b$ , is equal to $p^{t h}$ term of another GP, whose first term is $a$ and fifth term is $b$ . Then $p$ is equal to

Let $3, a, b, c$ be in A.P. and $3, a - 1, b + 1, c + 9$ be in G.P. Then, the arithmetic mean of $a, b$ and $c$ is :

If the 2nd, 5th and 9th terms of a non-constant A.P. are in G.P., then the common ratio of this G.P. is:

Three positive numbers form an increasing G. P. If the middle term in this G.P. is doubled, the new numbers are in A.P. then the common ratio of the G.P. is: