Problem 53
Question
Locus of mid point of the portion between the axes of \(x \cos \alpha+y \sin \alpha=p\) whre \(p\) is constant is [2002] (a) \(x^{2}+y^{2}=\frac{4}{p^{2}}\) (b) \(x^{2}+y^{2}=4 p^{2}\) (c) \(\frac{1}{x^{2}}+\frac{1}{y^{2}}=\frac{2}{p^{2}}\) (d) \(\frac{1}{x^{2}}+\frac{1}{y^{2}}=\frac{4}{p^{2}}\)
Step-by-Step Solution
Verified Answer
The correct locus is (d) \(\frac{1}{x^{2}} + \frac{1}{y^{2}} = \frac{4}{p^{2}}\).
1Step 1: Identify points on axes
The line equation given is \( x \cos \alpha + y \sin \alpha = p \). This equation can intersect with the x-axis when \( y = 0 \), giving us the x-intercept as \( (\frac{p}{\cos \alpha}, 0) \). Similarly, it intersects with the y-axis when \( x = 0 \), giving the y-intercept as \( (0, \frac{p}{\sin \alpha}) \). These are the endpoints.
2Step 2: Find midpoint coordinates
The midpoint of a segment is calculated by taking the average of the corresponding coordinates of its endpoints. Therefore, the midpoint \((m_x, m_y)\) is given by: \[ m_x = \frac{\frac{p}{\cos \alpha} + 0}{2} = \frac{p}{2 \cos \alpha} \] and \[ m_y = \frac{0 + \frac{p}{\sin \alpha}}{2} = \frac{p}{2 \sin \alpha} \]
Key Concepts
Coordinate GeometryConic SectionsJEE Main Mathematics
Coordinate Geometry
Coordinate geometry, also known as "Analytic Geometry," is a branch of mathematics where algebra is used to study geometric properties. It involves placing geometric figures such as points, lines, and circles in a coordinate plane and using equations to describe their properties. By assigning coordinates to each point, we can convert complex geometric problems into more manageable algebraic equations.
In the given exercise, the line equation involves using trigonometric functions and a constant. This approach is typical in coordinate geometry, where lines, angles, and distances are analyzed using coordinates. In this context, the line is expressed in terms of its relation to the axes using angle and distance, which allows us to find intercepts and midpoints easily. These concepts form the backbone of coordinate geometry, enabling us to explore various geometric relationships through calculation and visualization.
In the given exercise, the line equation involves using trigonometric functions and a constant. This approach is typical in coordinate geometry, where lines, angles, and distances are analyzed using coordinates. In this context, the line is expressed in terms of its relation to the axes using angle and distance, which allows us to find intercepts and midpoints easily. These concepts form the backbone of coordinate geometry, enabling us to explore various geometric relationships through calculation and visualization.
Conic Sections
Conic sections are shapes created by the intersection of a cone with a plane. The main types of conic sections are circles, ellipses, parabolas, and hyperbolas. These shapes show up in various physical phenomena and mathematical problems, making them an essential part of geometry and calculus.
In many problems involving conic sections, the goal is often to find specific properties, such as the equation of the shape or the locus of points defining it. The exercise discussed deals with this concept by asking for the locus of the midpoint of the segment defined by the intercepts of a line. While the solution focuses more on computing specific coordinates, the conceptual link to conic sections is present in understanding how these points relate to conic curves, possibly a circle or ellipse based on the locus derived.
In many problems involving conic sections, the goal is often to find specific properties, such as the equation of the shape or the locus of points defining it. The exercise discussed deals with this concept by asking for the locus of the midpoint of the segment defined by the intercepts of a line. While the solution focuses more on computing specific coordinates, the conceptual link to conic sections is present in understanding how these points relate to conic curves, possibly a circle or ellipse based on the locus derived.
- Circle: A simple form of conic defined by all points equidistant from a center point.
- Ellipse: Like a flattened circle, where the sum of the distances from any point on the ellipse to two foci is constant.
- Hyperbola: Formed by the difference in distances from any point on the hyperbola to two foci being constant.
JEE Main Mathematics
JEE (Joint Entrance Examination) Main Mathematics is a challenging component of an entrance exam for engineering colleges in India. It covers a wide range of mathematical topics, including algebra, calculus, and geometry. Students preparing for this exam need to have a robust understanding of these concepts.
Coordinate geometry and conic sections, as covered in this exercise, are part of the JEE Main syllabus. Problems often require students to find equations of lines, determine geometric loci, or understand the intersections of geometric shapes. To excel in JEE Main Mathematics, students will need to develop:
- Strong foundational knowledge in mathematical concepts.
- Problem-solving skills to apply these concepts in various contexts.
- Familiarity with complex equations and their geometric interpretations.
Other exercises in this chapter
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