Question

The ratio of the roots of the equation \(a x^{2}+b x+c=0\) is same as the ratio of the roots of the equation \(p x^{2}+q x+r=0\). If \(D_{1}\) and \(D_{2}\) are the discriminants of \(a x^{2}+b x+c=0\) and \(p x^{2}+q x+r=0\) respectively then \(D_{1}: D_{2}=\) (a) \(\frac{a^{2}}{q^{2}}\) (b) \(\frac{a^{2}}{p^{2}}\) (c) \(\frac{\dot{b}^{2}}{q^{2}}\) (d) none of these

Short answer

Question: Find the ratio of the discriminants of the quadratic equations \(ax^2 + bx + c = 0\) and \(px^2 + qx + r = 0\), where the same ratio of roots is found between the equations. Answer: (d) none of these

Step-by-step solution

Identify the quadratic equations and their discriminants

Given two quadratic equations are: 1. \(ax^2 + bx + c = 0\) with discriminant \(D_1\) 2. \(px^2 + qx + r = 0\) with discriminant \(D_2\) The discriminant, \(D\), of a quadratic equation given as \(ax^2 + bx + c = 0\) is: \(D = b^{2} - 4ac\) Therefore, we have: \(D_1 = b^{2} - 4ac\) for the first equation \(D_2 = q^{2} - 4pr\) for the second equation

Establish the relationships between the roots of both equations

Let the roots of the first quadratic \(ax^2 + bx + c = 0\) be \(\alpha\) and \(\beta\) and the roots of the second quadratic \(px^2 + qx + r = 0\) be \(\gamma\) and \(\delta\). According to the given problem, the ratio of the roots is the same for both equations: \(\frac{\alpha}{\beta} = \frac{\gamma}{\delta}\) From the properties of the roots of a quadratic equation, we know: \(\frac{\gamma}{\delta} = \frac{\alpha}{\beta} = \frac{ -(b/a)}{-(c/a)} = \frac{c}{b}\). Similarly, for the second equation: \(\frac{\gamma}{\delta} = \frac{ -(q/p)}{-(r/p)} = \frac{r}{q}\). Now, since the ratio of roots is same for both equations, we have: \(\frac{c}{b} = \frac{r}{q}\).

Compute the ratio of the discriminants

To find the ratio of the discriminants \(D_1 : D_2,\), we have: \(\frac{D_1}{D_2} = \frac{b^2 - 4ac}{q^2 - 4pr}\). From the relationship we determined in the previous step, \(\frac{c}{b} = \frac{r}{q}\), we can rewrite the ratio of discriminants as: \(\frac{D_1}{D_2} = \frac{b^2 - 4ac}{q^2 - 4(\frac{c}{b})q^2} = \frac{b^2 - 4ac}{q^2 - 4cq}\) Multiply numerator and denominator by \(b^2\) \(\frac{D_1}{D_2} = \frac{(b^2 - 4ac)b^2}{(q^2 - 4cq)b^2} = \frac{(b^2 - 4ac)b^2}{(q^2 - 4cq)b^2}\) Now, find the final form of the discriminant ratio and compare it with the options: (a) \(\frac{a^2}{q^2}\) (b) \(\frac{a^2}{p^2}\) (c) \(\frac{b^2}{q^2}\) (d) none of these

Compare the ratio with the options and select the correct answer

The given ratio of \(D_1 : D_2\) in its simplest form is \(\frac{(b^2 - 4ac)b^2}{(q^2 - 4cq)b^2}\). This does not match any of the options (a), (b), or (c). Therefore, the correct answer is: (d) none of these

Key concepts

Quadratic Equations

Quadratic equations are algebraic expressions that set a second-degree polynomial equal to zero, in the form of \(ax^2 + bx + c = 0\). Here, \(a\), \(b\), and \(c\) are constants, with \(a \eq 0\), and \(x\) represents the variable or unknown that we are trying to solve for.

They play a central role in mathematics and appear in various applications across physics, engineering, and economics. Understanding how to solve these equations is foundational for students, as they provide a basis for more advanced mathematical concepts. The solutions to a quadratic equation are known as its roots, which can be real or complex numbers.

The roots can be found using various methods such as factoring, completing the square, using the quadratic formula, or graphing. Each method is useful depending on the specific form and complexity of the equation presented.

Discriminant

The discriminant is a powerful tool in understanding the nature of the roots of a quadratic equation without explicitly calculating them. Given a quadratic equation \(ax^2 + bx + c = 0\), the discriminant \(D\) is calculated as \(D = b^2 - 4ac\).

The value of the discriminant gives us important information about the roots:

• If \(D > 0\), there are two distinct real roots.
• If \(D = 0\), there is exactly one real root, or the roots are real and equal (the quadratic touches the x-axis at a single point).
• If \(D < 0\), the roots are complex and conjugate to each other (the quadratic does not intersect the x-axis).

Understanding the discriminant allows students to quickly determine the number and type of solutions to a quadratic equation without having to find the actual roots.

Roots of Equation

The roots of a quadratic equation \(ax^2 + bx + c = 0\) are the values of \(x\) that make the equation true; that is, the values that satisfy the equation. These roots can be represented as \(x_1\) and \(x_2\) and can be determined by various solving techniques, including using the quadratic formula: \(x = \frac{-b \pm \sqrt{D}}{2a}\), where \(D\) is the discriminant.

Based on the properties of quadratic equations, the sum and product of the roots have specific relationships to the coefficients of the equation, which are: \(x_1 + x_2 = -\frac{b}{a}\) and \(x_1 \cdot x_2 = \frac{c}{a}\). These relationships are immensely helpful as they allow one to write an entirely new quadratic equation when provided with just the roots, or vice versa, to find the roots if the sum and product are known.

Ratio of Roots

In some cases, information regarding the ratio of the roots of quadratic equations can be given instead of their exact values. If the roots \(\alpha\) and \(\beta\) of the quadratic equation \(ax^2 + bx + c = 0\) have a ratio \(\alpha : \beta\), we can use the relationship between the roots and the coefficients to express this ratio as \(\frac{-b/a}{-c/a} = \frac{c}{b}\).

When comparing two quadratic equations with roots that have the same ratio, we can equate these ratios to find a relationship between the coefficients of the equations, which can be used to deduce further properties, such as the ratios of their discriminants. This concept is particularly useful when direct calculation of roots is complex or unnecessary, but the relative relationship between roots is what matters for a specific mathematical investigation or real-world application.