Problem 8 - A Complete Resource Book in Mathematics for JEE Main 2017

Question

If from a point, the two tangents drawn to the parabola \(y^{2}=4 a x\) are normals to the parabola \(x^{2}=4 b y\), then (A) \(a^{2}>8 b^{2}\) (B) \(b^{2}>8 a^{2}\) (C) \(a^{2}<8 b^{2}\) (D) none of these

Step-by-Step Solution

Answer

(C) \(a^{2}<8b^{2}\).

Step 1: Equation of Tangents to Parabola

For the parabola \(y^2 = 4ax\), if a point \((h, k)\) is on the tangents, the equation of tangents is given by \(yy_1 = 2a(x + x_1)\). The point \((x_1, y_1)\) is where the tangents are drawn from point \((h, k)\).

Step 2: Relation from Normality Condition to Second Parabola

For the parabola \(x^2 = 4by\), the slope of the normal at any point \((x_2, y_2)\) is \(-\frac{dy}{dx} = -\frac{x_2}{2b}\). Given that the tangents are normals to this parabola as well, they satisfy this slope condition.

Step 3: Slope Equations of Tangents and Normals

For the parabola \(y^2=4ax\), determine the slope of the tangent to be \(y_1/2a\). Substitute the coordinates of the normal from \(x^2=4by\) into this slope equation to relate the constants \(a\) and \(b\).

Step 4: Establish the Relationship between a and b

Using the normal slope condition and tangent equation, substitute and simplify the expressions \(-\frac{x_2}{2b} = \frac{y_1}{2a}\), leading to \(x_2 = \frac{b}{a}y_1\). Relate \(x_1\) and \(y_1\) using the parabola equation.

Step 5: Solving the Inequality

By solving the derived expressions through substitution, you obtain an equation that gives a condition on \(a^2\) and \(b^2\). Simplifying it will yield \(a^2 < 8b^2\) as the valid condition for the problem's scenario.

Key Concepts

Tangent equations

When working with parabolas, understanding tangent equations plays a crucial role. A tangent is a line that touches a curve at precisely one point, and for parabolas, these lines can be described using specific equations. For the parabola given by the equation \( y^2 = 4ax \), if a point \((h, k)\) lies on the tangents, the equation takes the form \(yy_1 = 2a(x + x_1)\). Here, \((x_1, y_1)\) are coordinates of the point where the tangent line intersects with the parabola. You can visualize it as a point gently brushing the side of the bowl-like shape of the parabola. Tangents have the unique property of having the same slope as the curve at the point of contact. This concept is extremely useful when you're trying to understand relationships between different curves, like in this exercise where tangents also serve as normals to another parabola.

Normal lines

Normal lines to a curve are lines perpendicular to the tangents at the point of contact. This fascinating geometrical property makes them instrumental in several calculations and solutions in geometry.For the parabola \(x^2 = 4by\), the slope of the normal at any point \((x_2, y_2)\) is calculated as \(-\frac{x_2}{2b}\). This is derived from the derivative of the curve at the point. This exercise indicates that the tangents from the other parabola \(y^2 = 4ax\) also function as normals to this one, meaning they share the slope condition.To grasp the role of normals, imagine standing on a hill. The tangent is like the direction you're facing, while the normal would be the steep path going straight downhill. Normal lines help in understanding the way surfaces curve and change direction and are pivotal in determining intersections and angles.

Inequalities in geometry

In geometry, inequalities help determine the relative sizes of segments, angles, or other parameters like constants in equations. In this exercise, you're working through an inequality involving the constants \(a\) and \(b\) of two different parabolas.The relationship between these constants is derived from conditions set within the tangents and normals. By solving through the tangent and normal equations, the crucial inequality \(a^2 < 8b^2\) is established.Inequalities in geometry serve many purposes. They help determine bounds and limits within a geometric setup, giving insight into the feasible values and solutions a geometrical construct can take. This helps in constructing, designing, and analyzing geometrical models, making the solution both practical and insightful.