Wednesday, March 22, 2023

# Analytic Geometry Circle Sample Problems

## Solution Of Exercise 3

Analytical Geometry (circles) Grade 12 (Further problems on Circles ) Part 2 Of 2

Calculate the equation of the circle that has its center at and has the y-axis as a tangent.

This time, the circle has the y-axis as a tangent. This means that the x coordinate will be zero. Hence, we have 2 coordinates which are C and T. We will use the distance formula again to find the value of the radius.

Putting the values of a, b, and r:

Finding the equation of the circle by using the standard equation:

## Compass And Straightedge Constructions

It is relatively straightforward to construct a line t tangent to a circle at a point T on the circumference of the circle:

• A line a is drawn from O, the center of the circle, through the radial point T
• The line t is the perpendicular line to a.

Thales’ theorem may be used to construct the tangent lines to a point P external to the circle C:

• A circle is drawn centered on the midpoint of the line segment OP, having diameter OP, where O is again the center of the circle C.
• The intersection points T1 and T2 of the circle C and the new circle are the tangent points for lines passing through P, by the following argument.

The line segments OT1 and OT2 are radii of the circle C since both are inscribed in a semicircle, they are perpendicular to the line segments PT1 and PT2, respectively. But only a tangent line is perpendicular to the radial line. Hence, the two lines from P and passing through T1 and T2 are tangent to the circle C.

Another method to construct the tangent lines to a point P external to the circle using only a straightedge:

• Draw any three different lines through the given point P that intersect the circle twice.
• Let

=r^} .

## Tangent Lines To Circles

In Euclidean plane geometry, a tangent line to a circle is a line that touches the circle at exactly one point, never entering the circle’s interior. Tangent lines to circles form the subject of several theorems, and play an important role in many geometrical constructions and proofs. Since the tangent line to a circle at a pointP is perpendicular to the radius to that point, theorems involving tangent lines often involve radial lines and orthogonal circles.

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## Tangent Lines To Two Circles

For two circles, there are generally four distinct lines that are tangent to both if the two circles are outside each other but in degenerate cases there may be any number between zero and four bitangent lines these are addressed below. For two of these, the external tangent lines, the circles fall on the same side of the line for the two others, the internal tangent lines, the circles fall on opposite sides of the line. The external tangent lines intersect in the external homothetic center, whereas the internal tangent lines intersect at the internal homothetic center. Both the external and internal homothetic centers lie on the line of centers , closer to the center of the smaller circle: the internal center is in the segment between the two circles, while the external center is not between the points, but rather outside, on the side of the center of the smaller circle. If the two circles have equal radius, there are still four bitangents, but the external tangent lines are parallel and there is no external center in the affine plane in the projective plane, the external homothetic center lies at the point at infinity corresponding to the slope of these lines.

The red line joining the points (

## Determine The Coordinates Of The Points On The Circle To calculate the possible coordinates of the point on the circle which have an \-value that is twice the \-value, we substitute \ into the equation of the circle:

This gives the points \\) and \\).

Note: we can check that both points lie on the circle by substituting the coordinates into the equation of the circle:

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## Mcq In Analytic Geometry: Points Lines And Circles Part 1

This is the Multiple Choice Questions Part 1 of the Series in Analytic Geometry: Points, Lines and Circles topics in Engineering Mathematics. In Preparation for the ECE Board Exam make sure to expose yourself and familiarize in each and every questions compiled here taken from various sources including but not limited to past Board Examination Questions in Engineering Mathematics, Mathematics Books, Journals and other Mathematics References.

#### MCQ Topic Outline included in Mathematics Board Exam Syllabi

• MCQ in Rectangular coordinates system | MCQ in Distance Formula | MCQ in Distance between two points in space | MCQ in Slope of a Line | MCQ in Angle between two lines| MCQ in Distance between a point and a line | MCQ in Distance between two lines | MCQ in Division of line segment | MCQ in Area by coordinates | MCQ in Lines | MCQ in Conic sections | MCQ in Circles

#### Start Practice Exam Test Questions Part 1 of the Series

Choose the letter of the best answer in each questions.

Problem 1: ECE Board April 1999

The linear distance between 4 and 17 on the number line is

A. 13

The equation of the circle is \^ + ^ = 40\).

 Circles with the same centre. Circles touch internally at one point: the distance between the centres is equal to the difference between the radii. Circles intersect at two points: the distance between the centres is less than the sum of the radii. Circles touch externally at one point: the distance between the centres is equal to the sum of the radii. Circles do not touch or intersect: the distance between the centres is greater than the sum of the radii.

## Equation Of A Circle With Centre At \\

Determine whether or not each of the following equations represents a circle. If not, give a reason.

Write down the equation of the circle:

Determine the centre and the length of the radius for the following circles:

Centre: \\), \ units

Centre: \\), \ units

Centre: \\), \ units

Centre: \\), \ units

Centre: \\), \ units

Centre: \\), \ units

Centre: \\), \ units

Centre: \\), \ units

A circle cuts the \-axis at \\) and \\). If \ units, determine the possible equation of the circle. Draw a sketch.

The equation of the circle passing through points \ and \ is \^ = 20\) or \^ = 20\).

\\) and \\) are points on a circle such that \ is a diameter. Determine the equation of the circle.

Use the distance formula to determine the length of the diameter.

Given \ is a diameter of the circle, then the centre is the mid-point of PG:

Therefore the centre is \\) and the equation of the circle is \^ + ^ = 25\).

A circle with centre \\) passes through the points \\) and \\).

The equation of the circle is \^ + ^ = 13\) .

Use the mid-point formula to calculate the coordinates of \:

Given that the centre of the circle lies on the line \, we can write the coordinates of the circle as \\) and the equation of the circle becomes:

The equation of the circle is \^ + ^ = 37\).

A circle with centre \\) passes through the point \\).

Centre of the shifted circle: \\)

Determine whether the circle \ cuts, touches or does not intersect the \-axis and the \-axis.

## Analytic Geometry In Three Dimensions

Analytic Geometry [Line Sample Problem]

In this, we consider triples which are real numbers and call this set as three- dimensional number space and denote it by R. All the elements in the triple are called coordinates.

Lets see how three-dimensional number space is represented on a geometric space.

In three-dimensional space, we consider three mutually perpendicular lines intersecting in a point O. these lines are designated coordinate axes, starting from 0, and identical number scales are set up on each of them.

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## What Is Analytic Geometry

Analytic geometry is that branch of Algebra in which the position of the point on the plane can be located using an ordered pair of numbers called as Coordinates. This is also called coordinate geometry or the Cartesian geometry. Analytic geometry is a contradiction to the synthetic geometry, where there is no use of coordinates or formulas. It is considered axiom or assumptions, to solve the problems. But in analytic geometry, it defines the geometrical objects using the local coordinates. It also uses algebra to define this geometry.

Coordinate geometry has its use in both two dimensional and three-dimensional geometry. It is used to represent geometrical shapes. Let us learn the terminologies used in analytic geometry, such as

• Plane
• Coordinates

## Solution Of Exercise 6

A triangle with vertices A = , B = and C = is inscribed in a circle. Calculate the equation of this circle.

We will insert all the coordinates in the standard equation to find the value of g, f, and c. Once we find the values of g,f, and c then we will insert all those values in the standard equation to develop the equation of the circle.

Plugging the coordinates of A:

Plugging the coordinates of B:

Plugging the coordinates of C:

After solving the above equations, we will get:

Putting all the values in the general equation of the circle:

## Tangent Lines To One Circle

A tangent line t to a circle Cintersects the circle at a single point T. For comparison, secant lines intersect a circle at two points, whereas another line may not intersect a circle at all. This property of tangent lines is preserved under many geometrical transformations, such as scalings, rotation, translations, inversions, and map projections. In technical language, these transformations do not change the incidence structure of the tangent line and circle, even though the line and circle may be deformed.

The radius of a circle is perpendicular to the tangent line through its endpoint on the circle’s circumference. Conversely, the perpendicular to a radius through the same endpoint is a tangent line. The resulting geometrical figure of circle and tangent line has a reflection symmetry about the axis of the radius.

The tangent line t and the tangent point T have a conjugate relationship to one another, which has been generalized into the idea of pole points and polar lines. The same reciprocal relation exists between a point P outside the circle and the secant line joining its two points of tangency.

If a point P is exterior to a circle with center O, and if the tangent lines from P touch the circle at points T and S, then TPS and TOS are supplementary .

If a chord TM is drawn from the tangency point T of exterior point P and PTM 90° then PTM = TOM.

## Solution Of Exercise 7 The ends of the diameter of a circle are the points A = and B = . What is the equation of this circle?

We will find the diameter with the help of the distance formula. To find the radius, we will divide the diameter into half. Furthermore, if we find the midpoint of the AB line, that will be the center of the circle. We will use the midpoint formula to find the coordinates of the center of the circle.

Since, midpoint is the center of the circle, therefore, we will declare midpoint as the center of the circle:

Plugging the value of C and r in the standard equation:

You can also find the equation from the general equation of the circle:

Find various Maths tutors on Superprof.

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## Equation Of A Circle With Centre At The Origin

• Draw a system of axes with a scale of \ unit on the \-axis and on the \-axis.
• Draw the lines \ and \.
• Plot the following points:
• \\)
• \\)
• \\)
• \\)
• \\)
• \\)
• \\)
• What object do the points form?

• Measure the following distances and complete the table below:
 Line segment \ \
• Use the distance formula to check the results of the above table: \
 Line segment
• What do you notice about the length of each line segment?
• What is the general term given to this type of line segment?
• If the point \\) lies on the circle, use the distance formula to determine an expression for the length of \.
• Can you deduce a general equation for a circle with centre at the origin?
• A circle is the set of all points that are an equal distance from a given point . In other words, every point on the circumference of a circle is equidistant from its centre.

The radius of a circle is the distance from the centre of a circle to any point on the circumference.

A diameter of a circle is any line passing through the centre of the circle which connects two points on the circle. The diameter is also the name given to the maximum distance between two points on a circle.

Consider a point \\) on the circumference of a circle of radius \ with centre at \\).

In \:

Equation of a circle with centre at the origin:

If \\) is a point on a circle with centre \\) and radius \, then the equation of the circle is:

Circle symmetry

## Determine If \ Is A Diameter Of The Circle

Since \ connects two points, \ and \, on the circle and is a line of length \ units, \ is a diameter of the circle.

Alternative method \: Use symmetry to show that \ is a diameter of the circle.

The diameter is the name given to the maximum distance between two points on a circle this means that the two points must lie opposite each other with respect to the centre.

Using symmetry about the origin, we know that \\) lies opposite \\) on the circle and vice versa:

• \\) lies opposite \\), which are the coordinates of \
• \\) lies opposite \\), which are the coordinates of \

Therefore, \ is a diameter of the circle.

Alternative method \: Show that \ passes through the centre.

Determine the equation of \ and show that it passes through the origin.

\\) lines on the line \.

Therefore, \ passes through the centre and is a diameter of the circle.

## Solution Of Exercise 2

Calculate the equation of the circle that has its center at and has the x-axis as a tangent.

In this question, the circle has an x-axis as a tangent. A tangent is a straight line that touches a curve. This means that the “y coordinate” will be zero. We will use the distance formula to find the radius. Currently, we have 2 coordinates which are: C, T

Plugging the values of C and T:

Since we know the value of the radius and center of the circle, the equation will be:

## An Interesting Application From Nature:

Recognizing points on a circle | Analytic geometry | Geometry | Khan Academy

The Nautilus Shell

The curves that we learn about in this chapter are calledconic sections. They arise naturally in many situations and are the result of slicing a cone at various angles.

Depending on where we slice our cone, and at what angle, we will either have a straight line, a circle, a parabola, an ellipse or a hyperbola. Of course, we could also get a single point, too.

## Math Exercises & Math Problems: Analytic Geometry Of The Conic Sections

Determine whether the given equation is an equation of the conic section. If so, identify the type of a conic section and its properties :

Determine the relative position of a straight line p and a circle k. If they have any intersection points, determine their coordinates :

Find the equation of a circle circumscribed about the triangle ABC, where A, B, C .

Find the equation of a circle passing through the points K, L, if the center of a circle lies on the straight line p: 2x + 3y 5 = 0.

Find the equation of an ellipse having the center at the origin of the coordinate system and passing through the points M and N.

Find the vertex equation of a parabola passing through the points A, B and C, if the axis of a parabola is parallel to the x-axis.

Find the equation of a hyperbola passing through the point H, if the asymptotes of a hyperbola are given by the equations y 3 = ±2.

Find the equation of an ellipse, if two of the vertices of an ellipse have coordinates C, D and the focus of an ellipse has coordinates F.

Find the vertex equation of a parabola passing through the point P, if the vertex of a parabola has coordinates V.

Find the equation of a hyperbola if the axis of a hyperbola is parallel to the x-axis, the center of a hyperbola is C, the semi-major axis of a hyperbola is a = and the eccentricity of a hyperbola is e = .

Find the equation of a circle whose diameter is a line segment AB, A, B.

## Why Study Analytic Geometry

Science and engineering involves the study of quantities that change relative to each other .

It is much easier to understand what is going on in these problems if we draw graphs showing the relationship between the quantities involved.

The study of calculus depends heavily on a clear understanding of functions, graphs, slopes of curves and shapes of curves. For example, in the Differentiation chapter we use graphs to demonstrate relationships between varying quantities.