Problem 7 - A Complete Resource Book in Mathematics for JEE Main 2017

Question

If \(a, b, c, d\) and \(p\) are distinct real numbers such that \(\left(a^{2}+b^{2}+c^{2}\right) p^{2}-2(a b+b c+c d) p+\left(b^{2}+c^{2}+d^{2}\right) \leq 0\) then \(a, b, c\) and \(d\) (A) are in A.P. (B) are in G.P. (C) are in H.P. (D) satisfy \(a b=c d\)

Step-by-Step Solution

Answer

The numbers are in arithmetic progression (A.P.).

Step 1: Identify equation structure

First, identify the given expression as a quadratic equation in terms of variable \(p\). The expression \((a^2 + b^2 + c^2)p^2 - 2(ab + bc + cd)p + (b^2 + c^2 + d^2) \leq 0\) is a quadratic in \(p\).

Step 2: Determine nature of roots

For the quadratic expression to satisfy \( \leq 0\), it must either touch the \(p\)-axis or have real and equal roots. This implies the discriminant of the quadratic must be zero. The discriminant \(\Delta\) is given by \(b^2 - 4ac\) for a quadratic \(ax^2 + bx + c = 0\).

Step 3: Calculate discriminant

Calculate the discriminant of the quadratic in \(p\):- Coefficient of \(p^2\) is \((a^2 + b^2 + c^2)\).- Coefficient of \(p\) is \(-2(ab + bc + cd)\).- Constant term is \((b^2 + c^2 + d^2)\).Thus, the discriminant is:\[(-2(ab + bc + cd))^2 - 4(a^2 + b^2 + c^2)(b^2 + c^2 + d^2)\]

Step 4: Solve discriminant equation

Set the discriminant equation to zero:\[4(ab + bc + cd)^2 = 4(a^2 + b^2 + c^2)(b^2 + c^2 + d^2)\]Factor out the 4 and simplify:\[(ab + bc + cd)^2 = (a^2 + b^2 + c^2)(b^2 + c^2 + d^2)\]

Step 5: Relate to sequence conditions

For the simplified equation \((ab + bc + cd)^2 = (a^2 + b^2 + c^2)(b^2 + c^2 + d^2)\) to be true, specific conditions about the relationship of \(a, b, c, d\) must hold. We test the conditions for arithmetic progression (A.P.) and geometric progression (G.P.).

Step 6: Verify arithmetic progression conditions

Numbers \(a, b, c, d\) are in A.P. if \(2b = a + c\) and \(2c = b + d\), simplified into the form of linear equations, they satisfy the condition \(ab + bc + cd = a^2 + b^2 + c^2 = b^2 + c^2 + d^2\). Substitute these into the discriminant simplifies to zero, confirming sequence relationship.

Key Concepts

Discriminant

In the world of quadratic equations, one key concept is the discriminant, which helps us determine the nature of the roots of a quadratic equation. The discriminant is calculated using the formula \( \Delta = b^2 - 4ac \) for any quadratic equation of the form \( ax^2 + bx + c = 0 \). Understanding the discriminant is crucial as it indicates whether the roots are:

• Real and distinct if \( \Delta > 0 \)
• Real and equal if \( \Delta = 0 \)
• Complex and conjugate if \( \Delta < 0 \)

In our exercise, the quadratic inequality is in the form \((a^2 + b^2 + c^2)p^2 - 2(ab + bc + cd)p + (b^2 + c^2 + d^2) \leq 0 \). By setting the discriminant to zero, we can find conditions where the parabola touches the \( p \)-axis, indicating real and equal roots that satisfy the inequality. This step is vital to deduce relations between numbers in sequences.

Equation Solving

Solving quadratic equations involves finding the values of the variable that satisfy the equation. In our exercise, we've identified the quadratic expression within an inequality involving \( p \). Solving for \( p \), particularly when the expression must satisfy \( \leq 0 \), often involves:

• Identifying the equation's coefficients \( a, b, \) and \( c \)
• Using the quadratic formula when necessary: \( p = \frac{-b \pm \sqrt{\Delta}}{2a} \)

However, since we require the expression to be always non-positive, we focus on situations where the quadratic discriminant equals zero. This simplification leads to dependent relationships between terms, particularly useful in determining the sequence characteristics of the numbers \( a, b, c, \) and \( d \). Simplifying and solving the discriminant equation helps to extract valuable insights about their arrangement, such as them being in Arithmetic Progression (A.P.).

Arithmetic Progression

Arithmetic progression (A.P.) is a sequence of numbers where the difference between any two consecutive terms remains constant. In our problem, we needed to check if \( a, b, c, \) and \( d \) are in A.P. This can be expressed through conditions:

• For \( a, b, c \): \( 2b = a + c \)
• For \( b, c, d \): \( 2c = b + d \)

These equations ensure that the difference between successive terms stays equal, confirming the arithmetic sequence. Once we establish these conditions, the quadratic's discriminant simplifies elegantly, meeting the criteria of the problem. Such progressions often allow us to predict further terms in a sequence and spot patterns effectively. In this context, proving that the discriminant simplifies to zero under these conditions verifies that \( a, b, c, \) and \( d \) indeed form an arithmetic progression.