Theorems based on Loci: Definition, Equation, Theorems, Examples

Theorems based on loci: A locus is a shape or curve formed by all points satisfying a given equation of the connection between coordinates or a point, line, or moving surface in mathematics. The locus defines all geometries as points, including circles, ellipses, parabolas, and hyperbolas. The word ‘locus’ is a Latin word which means “connecting places”. It describes the location of something.

The locus is when an object is placed somewhere or something happens in a specific area. Theorems based on loci give the definition of various geometrical and conical figures such as circles, parabolas, hyperbolas and ellipses. Let us learn more about the theorems of loci in this article.

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What are Locus of Points?

A set of all points whose location satisfies or is determined by one or more stated conditions is referred to as a locus of points in the geometry. In other words, the collection of points that satisfy a condition is referred to as the locus of a point satisfying that property. These shapes can be regular or asymmetrical. Locus is not used to describe shapes with vertices or angles inside them.

For example, a circle is a set of points in the plane equidistant from a given point, whereas a sphere is a set of points in three dimensions equidistant from a given point.

Theorems Based on Loci

There are six theorems for a locus in geometry.

Locus Theorem 1

The locus from point \(P\) to fixed distance \(d\) is defined as a circle with \(P\) as the centre and \(d\) as the radius.

We can use this theorem to find the region formed by joining all the points at the same distance from a single point.

Locus Theorem 2

The locus from line \(m\) at a fixed perpendicular distance \(d\) is defined as a pair of parallel lines located on either side of \(m\) at a distance \(d\) from line \(m\).

With the help of this theorem, we can find the region formed by all the points at the same distance from a single line.

Locus Theorem 3

The loci that are equidistant from two points, \(A\) and \(B\), is a perpendicular bisector of the line segment that connects the two points.

This theorem can be used to determine the region formed by the points which lies at the equal distance from point \(A\) and point \(B\).

Locus Theorem 4

The locus that is equidistant from two parallel lines says \({m_1}\) and \({m_2}\) are said to be a line parallel to both lines and must be halfway between them.

This theorem aids in determining the region formed by all points at the same distance from two parallel lines.

Locus Theorem 5

The bisector of an angle is a locus that exists interior of the angle and is equidistant from its sides.

This theorem is used to determine the region formed by equidistant points from the arms of the angle. The angle bisector should be the region.

Locus Theorem 6

The locus, which is equidistant from two intersecting lines, \({m_1}\) and \({m_2}\), are defined as a pair of lines which bisect the angle formed by the two intersecting lines, \({m_1}\) and \({m_2}\).

This theorem aids in determining the region formed by all points located at the same distance from the two intersecting lines. The formed region should be a pair of lines that bisect the formed angle.

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Steps to Find Locus

can be described most naturally and easily as loci. A circle, for example, is the set of points in a plane that are at a fixed distance \(r\) from a given point \(P\), which is the centre of a circle.

Problems involving describing a specific locus are frequently solved by explicitly finding equations for the coordinates of the locus’s points. Here is a step-by-step guide to locating the loci.

• 1st Step: Choose a coordinate system that makes computations and equations as simple as possible.
• 2nd Step: Express the given conditions mathematically using the coordinates \(x\) and \(y\).
• 3rd Step: Reduce the complexity of the resulting equations.
• 4th Step: Determine the shape created by the equations.

Examples of Locus

• Angle Bisector

An angle bisector is a collection of points that bisect an angle and are equally distant from two intersecting lines that form an angle.

• Perpendicular Bisector

The collection of points that bisects the line and is formed by joining two points that are equally distant from each other is known as a perpendicular bisector.

• Parabola

A parabola is a collection of points that are equally distant from a fixed point and a line. The locus represents the fixed point, and the line represents the parabola’s directrix.

• Ellipse

The ellipse is defined as a group of points that satisfy the condition that the sum of the distances between two focal points is fixed.

• Hyperbola

A hyperbola has two distinct focal points that are both equally distant from the semi-major axis’s centre. The collection of points meets the condition that the absolute value of the difference in distances between two given foci is constant.

• Circle

The locus formed by the set of all points that are a fixed distance from a fixed point is called circle. Here, the fixed point is referred to as the circle’s “centre”, and the fixed distance is referred to as the circle’s “radius”.

Solved Examples – Theorems based on Loci

Below are a few solved examples that can help in getting a better idea.

Q.1. The intercept cuts off the coordinate axes by a variable straight line that adds up to \(10\) units. Find the point whose locus divides internally the part of the straight line intercepted between the coordinate axes in the ratio \(2:3\).

Ans: Assume that the variable straight line intersects the \(x-\)axis at \(A(a,0)\) and the \(y-\)axis at any point \(B(0,b)\).
The line intercepted between the coordinate axes is denoted by \(AB\).
Assume that the point \((h,k)\) internally divides the line-segment \(A,B\) in the ratio \(2:3\).
Then we have,
\(h = \frac{{2 \times 0 + 3 \times a}}{{2 + 3}}\)
\(h = \frac{{3a}}{5}\)
\(a = \frac{{5h}}{3}\)
And, \(k = \frac{{2 \times b + 3 \times 0}}{{2 + 3}}\)
\(k = \frac{{2b}}{5}\)
\(b = \frac{{5k}}{2}\)
According to the question,
\(a + b = 10\)
\(\frac{{5h}}{3} + \frac{{5k}}{2} = 10\)
\(\frac{{10h + 15k}}{6} = 10\)
\(\frac{5}{6}(2h + 3k) = 10\)
\(2h + 3k = 12\)
Hence, the equation of the locus of \((h,k)\) is \(2x + 3y = 12\).

Q.2. Find the equation of the locus of a point \(P\) that is at a distance of \(4\) units from the point \((-3,2)\).

Ans: Let the point on the locus be \(P(x,y)\)
Given the distance of a point, \(P(x,y)\) from \(( – 3,2)\) is \(4\).
\(\sqrt {{{(x – ( – 3))}^2} + {{(y – 2)}^2}} = 4\)
\({(x + 3)^2} + {(y – 2)^2} = 4\)
\({x^2} + 6x + 9 + {y^2} – 4y + 4 = 16\)
\({x^2} + 6x + {y^2} – 4y + 13 = 16\)
\({x^2} + 6x + {y^2} – 4y + 13 – 16 = 0\)
\({x^2} + 6x + {y^2} – 4y – 3 = 0\)
Hence, the locus of the required points is \({x^2} + 6x + {y^2} – 4y – 3 = 0\)

Q.3. Calculate the locus of a point which moves two units above the \(x-\) axis at a constant distance.

Ans: A locus is a collection of all points whose location satisfies or is determined by one or more specified geometrical conditions.
We know that the locus of a point that moves \(^\prime {k^\prime }\) units above the \(x-\)axis is given by \(y=k\) and \(^\prime {k^\prime }\) units below the \(x-\)axis are given by \(y=-k\).
Here, the point is moving two units above the \(x-\)axis
So, the locus of the given point is \(y=2\).

Q.4. Calculate the locus of a point that is equidistant from the points \(( – 2,0)\) and \((3,2)\).

Ans: Let \(P(x,y)\) be the given point on the locus.
Given, \(P(x,y)\) is equidistant from the points \(( – 2,0)\) and \((3,2)\).
\(\sqrt {{{(x – ( – 2))}^2} + {{(y – 0)}^2}} = \sqrt {{{(x – 3)}^2} + {{(y – 2)}^2}} \)
Squaring on both sides
\({(x + 2)^2} + {y^2} = {(x – 3)^2} + {(y – 2)^2}\)
\({x^2} + 4x + 4 + {y^2} = {x^2} – 6x + 9 + {y^2} – 4y + 4\)
\({x^2} + 4x + 4 + {y^2} – {x^2} + 6x – 9 – {y^2} + 4y – 4 = 0\)
\(10x + 4y – 9 = 0\)
Therefore, the locus of a point which equidistant from \(( – 2,0)\) and \((3,2)\) is \(10x + 4y – 9 = 0\)

Q.5. Find the equation to the locus of \(P\) for all values of the coordinates of a moving point \(P(a\cos \theta ,b\sin \theta )\).

Ans: Let \(P(x,y)\) be the point of the given locus
So, \(x = a\cos \theta \) and \(y = b\sin \theta \)
Thus,
\(\frac{x}{a} = \cos \theta \) and \(\frac{y}{b} = \sin \theta \)
Then the locus is given by
\({\left( {\frac{x}{a}} \right)^2} + {\left( {\frac{y}{b}} \right)^2} = {\cos ^2}\theta + {\sin ^2}\theta \)
\(\frac{{{x^2}}}{{{a^2}}} + \frac{{{y^2}}}{{{b^2}}} = 1\)
Hence, the required locus is \(\frac{{{x^2}}}{{{a^2}}} + \frac{{{y^2}}}{{{b^2}}} = 1\)

Summary

A locus is a collection of all points whose location satisfies or is determined by one or more specified geometrical conditions. There are various forms of locus, such as the locus of points equidistant from a fixed point at a fixed distance is a circle. The locus of points equidistant from lines is a parallel line. The locus of points that lie equidistant from two points is the perpendicular bisector.

The locus of points with the same distance from the sides of an angle is the angle bisector. The locus of points at the same distance from two intersecting lines is pair of perpendicular lines. The various conic sections such as circles, parabolas, hyperbolas, and ellipses are the loci of the points satisfying certain conditions.

FAQs on Theorems based on Loci

Students might be having many questions with respect to the Theorems based on Loci. Here are a few commonly asked questions and answers.

Q.1. What is the locus of a point equidistant from a triangle’s vertices?
Ans: The circumcentre of a triangle is the only point that is equidistant from all of the triangle’s vertices.

Q.2. How do you describe a locus geometrically?
Ans: A locus is a set of all points (commonly, a line, a line segment, a curve, or a surface) whose location satisfies or is determined by one or more specified conditions in geometry.

Q.3. What is a locus of a point?
Ans: The locus of points in two-dimensional geometry is a curve or a line. In geometry, a shape is defined by the locus of points. Assume that the centre of all equidistant locations is a circle. Other shapes, such as an ellipse, parabola, and hyperbola, are defined by the locus of the points.

Q.4. What is the locus of a circle?
Ans: A locus is a group of points that meet criteria. A circle’s locus is a collection of points always equidistant from the circle’s centre.

Q.5. What are loci used for?
Ans: The loci method (loci is a Latin word that means “connecting places”) is a memory enhancement strategy that employs visualisations of familiar spatial environments to improve information recall.

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