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Chapter 11: Problem 123

Let \(\mathrm{O}\) be the vertex and \(\mathrm{Q}\) be any point on the parabola, \(\mathrm{x}^{2}=8 \mathrm{y}\). If the point \(\mathrm{P}\) divides the line segment OQ internally in the ratio \(1: 3\), then locus of \(\mathrm{P}\) is : (a) \(y^{2}=2 x\) (b) \(x^{2}=2 y\) (c) \(x^{2}=y\) (d) \(y^{2}=x\)

Short Answer

Expert verified
The locus of \( P \) is \( x^2 = 2y \) (option b).

Step by step solution

01

Understand the Parabola Equation

The given parabola is described by the equation \( x^2 = 8y \). From this, we can identify the vertex as \( O(0,0) \), which is at the origin.
02

Parameterize the Point Q

Consider a point \( Q \) on the parabola. This point can be parameterized as \( Q(t, \frac{t^2}{8}) \) since it must satisfy the parabola equation \( x^2 = 8y \).
03

Use Section Formula to Find Coordinates of P

The point \( P \) divides the line segment \( OQ \) internally in the ratio 1:3. Using the section formula, the coordinates of \( P \) are given by \( P = \left( \frac{1 \cdot t + 3 \cdot 0}{1+3}, \frac{1 \cdot \frac{t^2}{8} + 3 \cdot 0}{1+3} \right) = \left( \frac{t}{4}, \frac{t^2}{32} \right) \).
04

Find Relation Between Coordinates of P

In \( P \), we have \( x = \frac{t}{4} \) and \( y = \frac{t^2}{32} \). Express \( t \) from \( x = \frac{t}{4} \) as \( t = 4x \). Substitute \( t \) in \( y \): \( y = \frac{(4x)^2}{32} = \frac{16x^2}{32} = \frac{x^2}{2} \).
05

Write the Locus of P

The derived relationship \( y = \frac{x^2}{2} \) is the equation of the locus of \( P \). Therefore, the locus of \( P \) is described by the equation \( x^2 = 2y \).

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Parabola
A parabola is a U-shaped curve that is one of the core concepts in conic sections. It is defined as the set of all points that are equidistant from a fixed point called the focus and a line known as the directrix.
A standard form of the parabola's equation is either in the form of \( y^2 = 4ax \) or \( x^2 = 4ay \), depending on its orientation.
  • For a vertically oriented parabola like \( x^2 = 8y \), the vertex is typically at the origin, \( (0, 0) \), and this is where the curve is the narrowest.
  • The axis of symmetry is a vertical line through the vertex, allowing the parabola to be symmetrical.
Understanding parabolas is crucial as they often appear in problems involving motion, optics, and various optimization scenarios.
Locus
The term "locus" refers to the set of all points that satisfy a specific condition or a set of conditions.
In coordinate geometry, finding the locus is a common exercise where you'll derive an equation that represents all the positions a point can occupy given its path constraint.
  • In the context of the problem, the locus pertains to the path followed by point \( P \) as \( Q \) moves along the parabola.
  • By understanding the transformation and division of segments, we derive the locus as \( x^2 = 2y \).
This helps visualize dynamic systems where position changes according to set rules and conditions.
Section Formula
The section formula is a mathematical method used to find the coordinates of a point that divides a line segment into a certain ratio.
It is particularly useful in coordinate geometry.
  • If point \( P \) divides a line segment joining \( A(x_1, y_1) \) and \( B(x_2, y_2) \) in the internal ratio \( m:n \), then coordinates of \( P \) are given by:
\[ (x, y) = \left( \frac{mx_2 + nx_1}{m+n}, \frac{my_2 + ny_1}{m+n} \right) \]
This simple formula applies the concept of weighted averages to determine the location of dividing points on a segment. It's essential for figuring out scenarios like those in our problem, where we want to trace the locus of a point dividing a segment on a parabola.
Coordinate Geometry
Coordinate geometry, also known as analytic geometry, utilizes algebra to study geometric problems.
It allows the representation and analysis of geometric figures through algebraic equations.
  • This field links geometric shapes and algebraic expressions through the coordinate plane, using formulas and methodologies to solve problems related to positions and distances.
  • Coordinate geometry is indispensable when deriving the equation of a locus as it helps transition from a geometric configuration to a numerical solution.
This branch of mathematics is key for understanding and solving descriptions of points, lines, and curves, including parabola, correlating spatial understanding with algebraic operation.

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Most popular questions from this chapter

A variable circle passes through the fixed point \(A(p, q)\) and touches \(x\)-axis. The locus of the other end of the diameter through \(A\) is (a) \((y-q)^{2}=4 p x\) (b) \((x-q)^{2}=4 p y\) (c) \((y-p)^{2}=4 q x\) (d) \((x-p)^{2}=4 q y\)

Statement 1: The only circle having radius \(\sqrt{10}\) and a diameter along line \(2 x+y=5\) is \(x^{2}+y^{2}-6 x+2 y=0\). Statement \(\mathbf{2}: 2 x+y=5\) is a normal to the circle $x^{2}+y^{2}-6 x+2 y=0 . (a) Statement 1 is false; Statement 2 is true. (b) Statement 1 is true; Statement 2 is true, Statement 2 is a correct explanation for Statement 1 . (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 for Statement 1 .

If the line \(y=m x+1\) meets the circle \(x^{2}+y^{2}+3 x=0\) in two points equidistant from and on opposite sides of \(x\)-axis, then (a) \(3 m+2=0\) (b) \(3 m-2=0\) (c) \(2 m+3=0\) (d) \(2 m-3=0\)

All the points in the set \(\mathbf{S}=\left\\{\frac{\alpha+\mathrm{i}}{\alpha-1} ; \propto \in \mathrm{R}\right\\}(\mathrm{i}=\sqrt{-1})\) lie on a: (a) straight line whose slope is 1 . (b) circle whose radius is 1 . (c) circle whose radius is \(\sqrt{2}\). (d) straight line whose slope is \(-1\).

If the pair of lines \(a x^{2}+2(a+b) x y+b y^{2}=0\) lie along diameters of a circle and divide the circle into four sectors such that the area of one of the sectors is thrice the area of another sector then (a) \(3 a^{2}-10 a b+3 b^{2}=0\) (b) \(3 a^{2}-2 a b+3 b^{2}=0\) (c) \(3 a^{2}+10 a b+3 b^{2}=0\) (d) \(3 a^{2}+2 a b+3 b^{2}=0\)
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