Question

The locus of a point equidistant from a fixed point is a
A. Circle
B. Straight Line
C. Parabola
D. Hyperbola

Answer 1

Hint: The locus of a point is the trajectory or path traced by it along the given condition and to find it we must use the distance formula which is $Distance=\sqrt{{{\left( {{x}_{2}}-{{x}_{1}} \right)}^{2}}+{{\left( {{y}_{2}}-{{y}_{1}} \right)}^{2}}}$ where $\left( {{x}_{1}},{{y}_{1}} \right)$ is the given point and also the fixed point, and $\left( {{x}_{2}},{{y}_{2}} \right)$ is the general point that moves and traces a path or simply any point on the curve. We are going to derive the equation of the locus traced by this point's motion around a fixed point such that the distance between them is fixed on any and every point that lies on the curve.

Complete step-by-step answer:
First of all we need to understand what is the locus of a point, it is simply described as the path traced by the motion of the point (or trajectory) by following the conditions given in the question that need to be followed.
We are given with only one condition that the distance between the fixed point and the moving point, which traces the locus, is constant on every point on the curve. So, we are going to use the distance formula by putting the values of the coordinates of these two points and equate it to some constant.
Our main motive is to form an equation and analyze the curve that equation represents.
We need to memorize general equations of some curves before we proceed with the question.
The important curves include:
Line: $\left( y-{{y}_{1}} \right)=m\left( x-{{x}_{1}} \right)$, one point form is the most common one.
Parabola: $y=a{{x}^{2}}+bx+c$, also called quadratic polynomial.
Circle: ${{\left( x-h \right)}^{2}}+{{\left( y-k \right)}^{2}}={{r}^{2}}$, with center having coordinates $\left( h,k \right)$.
Ellipse: $\dfrac{{{\left( x-h \right)}^{2}}}{{{a}^{2}}}+\dfrac{{{\left( y-k \right)}^{2}}}{{{b}^{2}}}=1$, with center at $\left( h,k \right)$.
Hyperbola: $\dfrac{{{\left( x-h \right)}^{2}}}{{{a}^{2}}}-\dfrac{{{\left( y-k \right)}^{2}}}{{{b}^{2}}}=1$, with center at $\left( h,k \right)$.
Now proceeding with our question and applying Distance formula between the fixed given point and the moving point which traces locus, we get,
Distance formula, $Distance=\sqrt{{{\left( {{x}_{2}}-{{x}_{1}} \right)}^{2}}+{{\left( {{y}_{2}}-{{y}_{1}} \right)}^{2}}}$
Let us consider ${{x}_{1}}=h$ and ${{y}_{1}}=k$ because it is a fixed point so its coordinates are constant, and also ${{x}_{2}}=x$ and ${{y}_{2}}=y$ because it is a moving point which will trace the curve and will be at every points position lying on the resulting curve, so it is a general point on curve traced.
So, by applying distance formula, we get,
$Distance=\sqrt{{{\left( x-h \right)}^{2}}+{{\left( y-k \right)}^{2}}}$
As the distance is constant, let us name it $r$
So, $r=\sqrt{{{\left( x-h \right)}^{2}}+{{\left( y-k \right)}^{2}}}$
Now squaring both sides for simplicity as there is no equation of curve having square roots, we get,
${{r}^{2}}={{\left( x-h \right)}^{2}}+{{\left( y-k \right)}^{2}}$
Now resembling this equation with the chart above, that we need to remember, we can analyze that this derived equation is the same as that of circle and hence the locus traced by the moving point is of circle.
therefore the correct option is A. Circle

Note: One needs to remember the equations mentioned in the solution in order to solve these types of questions on locus. If one needs to find the locus of parabola then the conditions given will be different which will include the ratio of the distance from a fixed point and a fixed line is 1, or we can say that they both are equal, and is called eccentricity (e) of parabola. In the case of Ellipse the condition is that the sum of distances of one moving point from two fixed points is constant and so on for other curves. Do not get confused while analyzing the derived equations and comparing them with the memorized general equations, bring down your derived equation in the general terms learned then there will be no chances for mistakes.