Problem 134 - 71 JEE Main Mathematics Online (2020 - 2012) & Offline (2018 - 2002) Chapterwise + Topicwise Solved Papers

Question

Statement 1: \(y=m x-\frac{1}{m}\) is always a tangent to the parabola, \(y^{2}=-4 x\) for all non-zero values of \(m\). Statement 2: Every tangent to the parabola, \(y^{2}=-4 x\) will meet its axis at a point whose abscissa is non- negative. (a) Statement 1 is true, Statement 2 is true; Statement 2 is a correct explanation of Statement 1 . (b) Statement 1 is false, Statement 2 is true. (c) Statement 1 is true, Statement 2 is false. (d) Statement 1 is true, Statement 2 is true, Statement 2 is not a correct explanation of Statement 1 .

Step-by-Step Solution

Answer

(d) Statement 1 is true, Statement 2 is true; Statement 2 is not a correct explanation of Statement 1.

Step 1: Understand the given equations

Statement 1 involves the line equation \( y = mx - \frac{1}{m} \) and the parabola equation \( y^2 = -4x \). We start by noting that the parabola opens to the left due to the negative coefficient.

Step 2: Condition for tangency

For a line to be tangent to a parabola, let's equate the line equation with the parabola equation by substituting \( y = mx - \frac{1}{m} \) into \( y^2 = -4x \). This gives us:\[(mx - \frac{1}{m})^2 = -4x\]This condition must simplify to the standard tangent form.

Step 3: Simplification and checking tangency

Expanding and simplifying \( (mx - \frac{1}{m})^2 = -4x \), we get:\[ m^2 x^2 - 2x + \frac{1}{m^2} = -4x \]\[ m^2 x^2 - 2x + \frac{1}{m^2} + 4x = 0 \]\[ m^2 x^2 + 2x + \frac{1}{m^2} = 0 \]For this to represent a single intersection point (tangency), there should be a perfect square or linear term left, confirming tangency.

Step 4: Solve quadratic relation

Analyze the quadratic equation formed. The equation shows characteristics suggesting it maintains tangency for all non-zero \( m \). The condition simplifies to confirm tangency.

Step 5: Analyze Statement 2

Statement 2 asserts that a tangent of the form in Statement 1 intersects the parabola's axis (y-axis here since vertex is origin) at a non-negative x-coordinate. Substitute the y-intercept \( y = mx - \frac{1}{m} \) set as zero. Since \( x = -\frac{1}{m^2} \) for the y-intercept, the minimum value is zero as \( m \to \infty \), otherwise it's negative.

Step 6: Compare explanations

Notice that Statement 1 focuses purely on tangency conditions universally for given m, while Statement 2 involves specific geometry regarding axis intersections, it doesn’t explain why Statement 1 holds for any m. Check if one logically leads to or explains the other.

Key Concepts

Quadratic Equations

Quadratic equations are polynomial equations of the second degree, generally in the form of \( ax^2 + bx + c = 0 \). The solutions to these equations can be found using various methods, including factoring, completing the square, or utilizing the quadratic formula \( x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} \). These equations are fundamental in mathematics as they describe the parabolic behavior of graphs, like those seen in projectile motion or architectural designs. Each quadratic equation graphically represents a parabola, which can open upwards or downwards depending on the sign of the coefficient \( a \).
The roots or solutions of the quadratic equation determine the x-intercepts of the parabola, defining where it crosses the x-axis. Additionally, understanding how the vertex form \( y = a(x-h)^2 + k \) connects to these equations helps clarify the vertex position, symmetry, and the direction in which the parabola opens.

Axis of Parabola

The axis of a parabola is a vital concept, serving as the line of symmetry for the graph. For the parabola equation \( y^2 = -4x \), the axis is the vertical line \( x = 0 \), commonly known as the y-axis. This axis of symmetry is significant as it divides the parabola into mirror-image halves, offering a pathway to better understand the parabola's behavior.
The vertex of a parabola, which is either the highest or lowest point of the curve, lies on this axis. This makes the axis a critical component in determining the parabola's shape and orientation within a coordinate plane. In understanding tangency problems, it becomes essential to know where tangents intersect this axis, as this determines whether such intersections have certain properties like non-negative x-coordinates. This aspect is crucial especially when analyzing whether tangents meet specific conditions like those outlined in various mathematical problems, including in the analysis of Statement 2 of the original exercise.

Tangent Lines

Tangent lines are straight lines that touch a curve at a single point, without crossing it at that location. They are important in calculus as they represent the instantaneous rate of change or derivative of the curve at that point. For a line to be a tangent to a parabola, there must only be a single point of contact. This is characterized by the equation of the line and the parabola sharing exactly one solution, indicating a unique point of intersection.
In the context of the equation \( y = mx - \frac{1}{m} \) being tangent to \( y^2 = -4x \), examining their intersection involves ensuring no overlap beyond this point. By substituting the line equation into the parabola's equation to find the conditions for tangency, one can determine whether a tangent exists for specific values of \( m \), as seen in the problem's solution. Understanding tangents helps solve problems relating to slopes and orientations of lines touching parabolas, crucial for skills in both geometry and calculus.