Problem 31 - A Complete Resource Book in Mathematics for JEE Main 2017

Question

The condition that the parabolas \(y^{2}=4 a x\) and \(y^{2}=4 c(x\) \(-b\) ) have a common normal other than \(x\)-axis \((a, b, c\) being distinct positive real numbers) is (A) \(\frac{b}{a-c}<2\) (B) \(\frac{b}{a-c}>2\) (C) \(\frac{b}{a-c}<1\) (D) \(\frac{b}{a-c}>1\)

Step-by-Step Solution

Answer

(A) \(\frac{b}{a-c}<2\)

Step 1: Identify the Common Normal

For two parabolas to have a common normal, the slopes of the normals at the points of the two parabolas must be the same. We are given the parabolas: \(y^2 = 4ax\) and \(y^2 = 4c(x-b)\). They also both might have a normal parallel to each other.

Step 2: Equation of Normal for Parabola

For a parabola \(y^2 = 4ax\), the equation of normal is derived from the slope \(-t\) of the tangent line at point \((at^2, 2at)\). Hence, the equation is: \(y = -tx + 2at + at^3\). For \(y^2 = 4c(x-b)\), assuming point \( (ct^2, 2ct) \), the normal is: \(y = -t(x-b) + 2ct + ct^3\).

Step 3: Equate Slopes of Parabolas

Set the slope of the normal for both parabolas equal at their respective points, getting the condition \(-t = -s\), implying that \(t = s\).

Step 4: Calculate Substituted Expressions

For the equation \(2at + at^3 = 2cs + cs^3 - bs\), and since \(t = s\), substitute \(s\) for \(t\) to solve as \(2a + at^2 = 2c + ct^2 - b\).

Step 5: Simplification

Rearranging gives \((a-c)t^2 + (2a - 2c) = -b\). For a non-trivial normal, set there the expression such that \(-b > (a-c + 2a - 2c)\). Simplifying further gives \(b < (a-c)(t^2 + 2)\), implying \(\frac{b}{a-c}<2\).

Step 6: Conclusion from Condition

The derived condition \(\frac{b}{a-c}<2\) implies that for these parabolas to have a non-trivial common normal other than the x-axis, this condition must be satisfied.

Key Concepts

Common Normal

A common normal is a line that acts as a perpendicular to the tangents of two parabolas at certain points. Imagine drawing a single straight line that intersects both parabolas and forms right angles with their tangent lines at the contact points. This line would be the common normal. For two parabolas to share a common normal, the key factor is that the slopes at these tangent points must be identical.
In the given problem, the task is to determine the condition under which the parabolas \( y^2 = 4ax \) and \( y^2 = 4c(x-b) \) have such a common normal. Ensuring the slopes of these tangent points justifies that a common line can serve as a normal.

Equation of Normal

The equation of a normal to a parabola represents the line that is perpendicular to its tangent at a point. For a parabola expressed by \( y^2 = 4ax \), at the point \( (at^2, 2at) \), the tangent has a slope of \(-t\). Hence, the normal equation becomes \( y = -tx + 2at + at^3 \).
Similarly, consider the parabola \( y^2 = 4c(x-b) \). At the point \( (ct^2, 2ct) \), its normal can be written as \( y = -t(x-b) + 2ct + ct^3 \). Deriving these normals is crucial to finding the specific line equations that could potentially be the common normal to both parabolas.

Slope of Tangents

The slope of the tangent is a measure of how steep or flat a tangent line is at a specific point on a curve. For parabolas like \( y^2 = 4ax \), the slope of the tangent at \( (at^2, 2at) \) is \(-t\). Similarly, for \( y^2 = 4c(x-b) \), the slope at \( (ct^2, 2ct) \) is also \(-t\).
This identical slope, \(-t\), means that the tangent lines at these points are perfectly equal in direction and steepness, allowing a common normal line to exist perpendicular to them. Verifying these slopes ensures compatibility between the tangents for establishing a common normal.

Quadratic Equation

Quadratic equations frequently arise when solving problems related to parabolas, including the condition of a common normal. When determining whether a common normal other than the x-axis exists, setting the expressions derived from the normal equations for equality leads to a quadratic equation.
For example, substituting \( s = t \) into the equation leads to a simplified condition: \( \( (a-c)t^2 + (2a - 2c) = -b \) \). The quadratic form here helps identify the relationships between the parameters \( a, b, \) and \( c \). It assists in determining the inequality \( \frac{b}{a-c}<2 \), guiding us to the solution's conclusion that ensures a non-trivial common normal.