Problem 131 - 71 JEE Main Mathematics Online (2020 - 2012) & Offline (2018 - 2002) Chapterwise + Topicwise Solved Papers

Question

Statement-1: The line \(x-2 y=2\) meets the parabola, \(y^{2}+2 x=0\) only at the point \((-2,-2)\). Statement-2: The line \(y=m x-\frac{1}{2 m}(m \neq 0)\) is tangent to the parabola, \(y^{2}=-2 x\) at the point \(\left(-\frac{1}{2 m^{2}},-\frac{1}{m}\right)\). (a) Statement- 1 is true; Statement- 2 is false. (b) Statement- 1 is true; Statement- 2 is true; Statement-2 is a correct explanation for statement- 1 . (c) Statement- 1 is false; Statement- 2 is true. (d) Statement-1 a true; Statement-2 is true; Statement-2 is not a correct explanation for statement-1.

Step-by-Step Solution

Answer

(d) Statement-1 is true; Statement-2 is true; Statement-2 is not a correct explanation for statement-1.

Step 1: Verify if the line meets the parabola at only one point

Substitute the value of \(x\) from the line equation \(x - 2y = 2\), i.e., \(x = 2y + 2\), into the parabola equation \(y^2 + 2x = 0\). This results in \(y^2 + 2(2y + 2) = 0\). Simplify to get \(y^2 + 4y + 4 = 0\), which factors into \((y+2)^2 = 0\). Hence, \(y = -2\). Substitute back to find \(x = 2(-2) + 2 = -2\). Thus, the point \((-2, -2)\) is the only intersection point, confirming Statement 1 is true.

Step 2: Verify tangency condition for the tangent to the parabola

Check the tangency condition given in Statement 2. For the given parabola \(y^2 = -2x\) and the line \(y = mx - \frac{1}{2m}\), substitute \(x\) from the line into the parabola to get \(y^2 = -2(mx - \frac{1}{2m})\). Rearranging gives \(y^2 = -2mx + \frac{1}{m}\). By substituting \(y = \frac{-1}{m}\) and \(x = -\frac{1}{2m^2}\) into the equation, the equation holds true and confirms tangency. Thus, Statement 2 is true.

Step 3: Analyze the relationship between the statements

Statement 2 provides information about a line being tangent to a parabola, which does not directly relate or explain the intersection of a different line with a different parabola as described in Statement 1. Thus, while both statements are true, Statement 2 is not a correct explanation for Statement 1.

Key Concepts

Coordinate Geometry

Coordinate Geometry is an essential branch of mathematics used to study the layout of points, lines, and shapes in a plane. It bridges algebra and geometry through graphical representation. By using a coordinate system, various geometric shapes can be represented with equations. In two-dimensional space, points are denoted by coordinates \(x, y\) which describe their position relative to the origin on a Cartesian plane.

Coordinate Geometry allows us to:

• Find distances between points.
• Determine the midpoint of a segment.
• Calculate the slope of a line.
• Identify symmetries of shapes.

Lines and curves, like parabolas and circles, are expressed using algebraic equations which can be systematically analyzed to identify intersections and other characteristics.

Parabola

A parabola is a simple but intriguing curve that is the graph of a quadratic function, often represented as \(y^2 = 4ax\) or \(x^2 = 4ay\). In this exercise, the parabolas \(y^2 + 2x = 0\) and \(y^2 = -2x\) depict downward and leftward opening curves, respectively. Parabolas have several important properties:

• The vertex is the point where the parabola changes direction. For \(y^2 = -2x\), it is at the origin.
• The axis of symmetry is a line that passes through the vertex, with the property that the parabola is mirrored across it.
• It involves a focal point and a directrix, both fundamental in defining its shape.

These characteristics make parabolas relevant in various real-world applications including paths of projectiles and satellite dishes.

Tangency Condition

A tangent line to a curve at any given point touches the curve without crossing it, much like how a car tire meets the road surface. When analyzed algebraically, this tangency condition can be checked by ensuring the intersection of the curve and the line results in exactly one unique solution. For parabolas, the condition for tangency involves equating the derivative of the parabola to the slope of the line, ensuring the orientations are matched.

Here, for the statement \(y = mx - \frac{1}{2m}\) being tangent to \(y^2 = -2x\), the point \(-\frac{1}{2m^2}, -\frac{1}{m}\) satisfies both the equations. The procedure involves:

• Substituting the line equation into the parabola's equation.
• Simplifying to verify the result holds true, indicating tangency.

Intersection Points

Intersection points occur when two curves or lines meet at one or more points in the Cartesian plane. Determining where a line intersects a parabola is a common problem in coordinate geometry.

The process involves substituting variables from one equation into another to find common solutions. In the case of the exercise:

• Substitute \(x = 2y + 2\) back into the parabola \(y^2 + 2x = 0\).
• Solve for \(y\) which leads to \(y = -2\).
• Find corresponding \(x\) to confirm the point \((-2, -2)\) as the only intersection point.

This meticulous procedure not only affirms correct solutions but also reveals whether lines merely touch or pass through a parabola's boundary.