Problem 101 - 71 JEE Main Mathematics Online (2020 - 2012) & Offline (2018 - 2002) Chapterwise + Topicwise Solved Papers

Question

If \(y=m x+4\) is a tangent to both the parabolas, \(y^{2}=4 x\) and \(x^{2}=2 b y\), then \(b\) is equal to: (a) \(-32\) (b) \(-64\) (c) \(-128\) (d) 128

Step-by-Step Solution

Answer

The value of \( b \) is \(-128\).

Step 1: Understand the relationships

First, recall that the line \( y = mx + c \) is tangent to a parabola when the quadratic equation formed by substitution has a discriminant of zero.

Step 2: Substitute in the first parabola

The given line is \( y = mx + 4 \). Substitute this equation into the first parabola \( y^2 = 4x \).\[ (mx + 4)^2 = 4x \] Simplify to form a quadratic in terms of \( x \).\[ (mx + 4)^2 = 4x \Rightarrow m^2x^2 + 8mx + 16 = 4x \]\[ m^2x^2 + (8m - 4)x + 16 = 0 \] The discriminant \( b^2 - 4ac \) should be zero: \[ (8m-4)^2 - 4(m^2)(16) = 0 \].

Step 3: Calculate the discriminant for the first parabola

Solve the equation \[ (8m-4)^2 - 64m^2 = 0 \]. Simplify and solve to find \( m \).\[ (8m-4)^2 = 64m^2 \] \[ 64m^2 - 64m + 16 = 64m^2 \] \[ -64m + 16 = 0 \] \[ m = \frac{1}{4} \].

Step 4: Substitute in the second parabola

Now, substitute \( y = mx + 4 \) into the second parabola \( x^2 = 2by \).\[ x^2 = 2b(mx + 4) \]\[ x^2 = 2b(mx) + 8b \] Rearrange and form a quadratic equation in terms of \( y \): \[ x^2 - 2bmx - 8b = 0 \]. The discriminant again should be zero.

Step 5: Condition for tangency for the second parabola

Apply the condition of tangency \( b^2 - 4ac = 0 \) to this quadratic equation: \[ (2bm)^2 - 4(1)(-8b) = 0 \]. Use \( m = \frac{1}{4} \) in the equation and solve for \( b \).\[ (2b \times \frac{1}{4})^2 + 32b = 0 \] \[ \frac{b^2}{4} + 32b = 0 \] \[ b^2 + 128b = 0 \].

Step 6: Solve for b

Factor out \( b \) from the equation: \[ b(b + 128) = 0 \]. So, either \( b = 0 \) or \( b = -128 \). Since \( b eq 0 \) (as the parabola equation \( x^2 = 2by \) would not make sense), the value is \( b = -128 \).

Step 7: Verify the choice

From the potential choices, verify that \( b = -128 \) is consistent with the answer choices provided in the question. Thus \( b = -128 \).

Key Concepts

Tangent Line

A tangent line is a straight line that touches a curve at exactly one point. This unique feature means that at the point of contact, both the tangent line and the curve have the same slope. For parabolas, which are U-shaped curves, a tangent is a line that meets the parabola at just one point, not crossing it.

In mathematical terms, if a line is given by the equation \( y = mx + c \), it will be tangent to a parabola if, when substituted into the parabola's equation, the resulting quadratic equation has its discriminant equal to zero. The discriminant is a measure that helps us understand the nature of the roots of the quadratic equation.

• If the discriminant is zero, the quadratic equation has exactly one solution, meaning the line just touches the parabola without crossing it.
• A positive discriminant indicates two points of intersection (the line crosses the parabola), while a negative discriminant implies no intersection.

Understanding the concept of a tangent is crucial for solving problems involving parabolas touching lines, as it allows us to determine specific conditions the line must satisfy.

Discriminant

The discriminant is an essential part of the quadratic formula. It determines the nature and number of solutions of a quadratic equation. When dealing with quadratic equations in the standard form \( ax^2 + bx + c = 0 \), the discriminant \( \Delta \) is given by the formula \( \Delta = b^2 - 4ac \).

This small but powerful value tells us:

• If \( \Delta > 0 \), we get two distinct real solutions.
• If \( \Delta = 0 \), we have exactly one real solution, indicating the line is tangent to the parabola at one point.
• If \( \Delta < 0 \), there are no real solutions, and the line does not intersect the parabola.

In the context of the exercise, we used the discriminant to ensure the tangency condition was met for the line and the parabola equations. By ensuring the discriminant equals zero, we mathematically confirm that the line is tangent to the parabola, touching it precisely at one point.

Quadratic Equation

A quadratic equation is a polynomial equation of degree two. It follows the general form \( ax^2 + bx + c = 0 \), where \( a \), \( b \), and \( c \) are constants, and \( a eq 0 \). These equations are vital in mathematics because they frequently describe various natural phenomena and geometric shapes, such as parabolas.

To solve a quadratic equation, we often use methods such as factoring, completing the square, or the quadratic formula. The quadratic formula being:\[x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}.\]This formula not only provides the solutions for \( x \) but also incorporates the discriminant, \( b^2 - 4ac \), which indicates the type of solutions we should expect.

• Two distinct solutions if the discriminant is positive.
• One solution if it's zero, suggesting a tangent case.
• No real solutions if negative.

Quadratic equations appear frequently in problems involving parabolas and their properties. In solving the original exercise, formulating the correct quadratic equation helped determine the position and interaction of the tangent with the parabola.