Problem 17 - A Complete Resource Book in Mathematics for JEE Main

Question

The two tangents to the curve \(a x^{2}+2 h x y+b y^{2}=1\), \(a>0\) at the points where it crosses \(x\)-axis, are (A) parallel (B) perpendicular (C) inclined at an angle \(\frac{\pi}{4}\) (D) None of these

Step-by-Step Solution

Answer

The tangents are parallel.

Step 1: Find Points of Intersection with the x-Axis

The curve is given by \(ax^2 + 2hxy + by^2 = 1\). To find points of intersection with the x-axis, we set \(y = 0\), leading to \(ax^2 = 1\). Solving for \(x\), we find \(x = \pm \frac{1}{\sqrt{a}}\). Thus, the points are \(\left( \frac{1}{\sqrt{a}}, 0 \right)\) and \(\left( -\frac{1}{\sqrt{a}}, 0 \right)\).

Step 2: Determine the Tangents to the Curve

To find tangents at these points, use implicit differentiation on \(ax^2 + 2hxy + by^2 = 1\). Differentiating with respect to \(x\), the equation becomes \(2ax + 2h(x \frac{dy}{dx} + y) + 2by \frac{dy}{dx} = 0\). Solving for \(\frac{dy}{dx}\) gives \(\frac{dy}{dx} = -\frac{ax + hy}{hx + by}\).

Step 3: Evaluate the Slope of Tangents at Intersection Points

Evaluate the slopes at the points where the curve intersects the x-axis. For \(y = 0\), the slope \(\frac{dy}{dx}\) becomes \(\frac{dy}{dx} = -\frac{a}{h}\). At both points \(\left( \frac{1}{\sqrt{a}}, 0 \right)\) and \(\left( -\frac{1}{\sqrt{a}}, 0 \right)\), the slope is \(-\frac{a}{h}\), meaning the tangents are parallel.

Key Concepts

Tangents to curves

Understanding the concept of tangents to curves is crucial for solving problems involving intersections with axes. A tangent to a curve at a particular point is a straight line that just "touches" the curve at that specific point. It reflects the direction in which the curve is heading at that point.

When a curve intersects an axis, the tangent lines at these points provide insights into its orientation and behavior.

• Slope of the Tangent: The slope of a tangent line is a measure of how steep the line is. At the point of tangency, this slope is equal to the derivative of the curve's equation at that point.
• Finding Tangents: To find a tangent, differentiate the curve's equation and solve for the slope at the desired point.

In this specific exercise, the curve intersects the x-axis at points where the value of \(y\) is zero. By employing implicit differentiation, we calculated the slopes of these tangents. The consistency of the calculated slopes reveals the direction and parallelism of the lines.

Implicit differentiation

Implicit differentiation is a technique used when you have an equation involving two variables where it isn't straightforward, or possible, to solve for one variable in terms of the other. Instead of expressing one variable explicitly in terms of the other, you differentiate both sides of the equation with respect to one variable, treating the other as a function of that variable.

This approach is particularly useful when dealing with complex curves not easily subjected to explicit differentiation. Here's how implicit differentiation works:

• Differentiation Process: Take the derivative of each term in the equation with respect to the desired variable (usually \(x\)). For any term involving \(y\), use the chain rule to differentiate, giving rise to expressions involving \(\frac{dy}{dx}\).
• Solve for \(\frac{dy}{dx}\): After differentiating, rearrange the terms to solve for \(\frac{dy}{dx}\), which is the derivative we're seeking.

In the given exercise, implicit differentiation allows us to find the slope of the tangent to the curve \(ax^2 + 2hxy + by^2 = 1\) by treating \(y\) as a function of \(x\) and differentiating accordingly. This process enables us to determine the behavior of the curve at any particular point.

Parallellism of lines

Parallel lines are lines in a plane that never intersect. They remain the same distance apart over their entire length. One of the most straightforward indicators of parallel lines is that they have the same slope. In geometry, recognizing parallelism helps understand the spatial arrangement and symmetry within figures.

When working with curves, finding parallel tangents means identifying points along the curve where the tangents share identical slopes. In terms of practical application:

• Equal Slopes: If two lines are parallel, they share the same slope. For example, a slope \(m\) of one tangent is equal to the slope of another tangent \(m\).
• Geometric Arrangement: Parallel lines in the context of curves indicate a sort of symmetry or repeated behavior at different sections along the curve.

In the original exercise, the slopes of tangents at the points of intersection with the x-axis were calculated to be \(-\frac{a}{h}\) at both points. This equality in slope confirms that the tangents are indeed parallel. Understanding this helps visualize how the curve intersects the axis and the orientation of the tangents at those intersections.