Question

Let \[\left( {{\mathbf{x}},{\mathbf{y}}} \right)\] be any point on the parabola \[{{\mathbf{y}}^{\mathbf{2}}} = {\mathbf{4x}}\]. Let P be the point that divides the line segment from \[\left( {{\mathbf{0}},{\mathbf{0}}} \right)\] and \[\left( {{\mathbf{x}},{\mathbf{y}}} \right)\] in the ratio \[{\mathbf{1}}:{\mathbf{3}}\]. Then the locus of P is :
A) ${x^2} = y$
B) ${y^2} = 2x$
C) ${y^2} = x$
D) ${x^2} = 2y$

Answer 1

Hint: We will assume the coordinates of P and using section formula we will find coordinates of p in terms of x and y in the equation. Now substituting the coordinates of P in the equation of parabola we will find the locus of P.

Complete step-by-step answer:
Let the coordinates of P, whose locus is to be determined to be \[\left( {h,k} \right)\].
Since we need to find the point which divides the line segment internally in the ratio 1 : 3, we have to use the section formula
Since this point divides the line joining \[\left( {{\mathbf{0}},{\mathbf{0}}} \right)\] to \[\left( {{\mathbf{x}},{\mathbf{y}}} \right)\] in the ratio\[{\mathbf{1}}:{\mathbf{3}}\]. So coordinates of point P will be \[(\dfrac{{x}}{4},\dfrac{y}{4})\]
So, \[h = \dfrac{x}{4}\] ​ and \[k = \dfrac{y}{4}\] (Using section formula)
Or, \[x = 4h\], and \[y = 4k\]
Since, \[{y^2} = 4x\]
$ \Rightarrow {\left( {4k} \right)^2} = 4\left( {4h} \right)$
\[ \Rightarrow \;{k^2} = h\]
Or \[{y^2} = x\], which is the locus of P.

So, option (C) is the correct answer.

Additional Information: A parabola may be defined as the locus of a point \[P\left( {x,{\text{ }}y} \right)\] whose distance from a given fixed point equals its distance from a given fixed line. The fixed point is known as the focus and the fixed line is known as the directrix. The vertex is the minimum or maximum point of the parabola. The axis of symmetry is a line which bisects the parabola. The focal length is the distance between the vertex and the focus or the vertex and the directrix.
The locus of the parabola is of the simplest non-degenerate conic sections to study. The definition of a parabola is the locus of a point Latex formula which moves such that its distance from a given point (called the focus) is equal to its perpendicular distance from a line (called the directrix). We can derive the most commonly encountered equation of the parabola given this definition.

Note: The coordinates of the point \[P\left( {x,{\text{ }}y} \right)\] which divides the line segment joining the points \[A\left( {{x_1},{\text{ }}{y_1}} \right)\] and \[B\left( {{x_2},{\text{ }}{y_2}} \right)\], internally, in the ratio \[{m_1}{\text{ }}:{\text{ }}{m_2}\]are \[\left\{ {\dfrac{{\left( {{m_1}{x_2} + {\text{ }}{m_2}{x_1}} \right)}}{{\left( {{m_1} + {\text{ }}{m_2}} \right)}}{\text{ }},\;\dfrac{{\left( {{m_1}{y_2} + {\text{ }}{m_2}{y_1}} \right)}}{{\left( {{m_1} + {\text{ }}{m_2}} \right)}}} \right\}\]. This is known as the section formula.