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Conic Sections: Problems with Solutions
Circles and Conic Sections
A series of free, online video lessons with examples and solutions to help Algebra students learn about circle conic sections.
Related Pages Conic Sections: Circles Conic Sections: Ellipses Conic Sections: Parabolas Conic Sections: Hyperbolas
What Is The Standard Form Of The Equation Of A Circle?
A circle is a set of points (x, y) which are a fixed distance r, the radius, away from a fixed point (h, k), the center. The equation of the circle in standard form is (x - h) 2 + (y - k) 2 = r 2
If a circle is given in general form then we must complete the square on the x and y parts of the equation to rewrite it in standard form.
Conic Sections: The Circle
In this video, you will learn how to identify the equation of a circle, how to write the standard form of a circle from the general form and how to graph a circle.
Conic Sections - Circle Part 1 of 2
Intuitive Math Help - Circles
Conic Sections - Circle Part 2 of 2
Circles (Intuitive Math Help)
Writing Equations Of Circles
This video covers writing equations of circles given information like the radius, center, or a point the circle goes through and the center.
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PRACTICE PROBLEMS ON CIRCLES IN CONIC SECTIONS
Problem 1 :
The equation of the circle passing through (1, 5) and (4, 1) and touching y-axis is
x 2 + y 2 - 5x - 6y + 9 + λ(4x+3y-19) = 0
where λ is equal to
a) 0, -40/9 b) 0 3) 40/9 d) -40/9
Problem 2 :
The circle x 2 + y 2 = 4x + 8y + 5 intersects the line 3x - 4y = m at two distinct points if
a) 15 < m < 65 b) 35 < m < 85
3) -85 < m < -35 d) -35 < m < 15
Problem 3 :
The length of the diameter of the circle which touches the x-axis at the point (1, 0) and passes through the point (2, 3).
Problem 4 :
The radius of the circle 3x 2 + by 2 + 4bx - 6by + b 2 = 0
a) 1 b) 3 c) √10 d) √11
Problem 5 :
The center of the circle inscribed in a square formed by the lines x 2 - 8x + 12 = 0 and y 2 - 14y + 45 = 0 is
a) (4, 7) b) (7, 4) c) (9, 4) d) (4, 9)
Problem 6 :
The equation of the normal to the circle x 2 + y 2 - 2x - 2y + 1 = 0 which is parallel to the line 2x + 4y = 3 is
a) x + 2y = 3 b) x + 2y + 3 = 0
c) 2x + 4y + 3 = 0 d) x - 2y + 3 = 0
Problem 7 :
The radius of the circle is passing through the point (6, 2) two of whose diameters are x + y = 6 and x + 2y = 4 is
a) 10 b) 2 √5 c) 6 d) 4
Problem 8 :
The equation of the circle passing through the foci of the ellipse
having center at (0, 3) is
a) x 2 + y 2 - 6y - 7 = 0 b ) x 2 + y 2 - 6y + 7 = 0
c) x 2 + y 2 - 6y - 5 = 0 d) x 2 + y 2 - 6y + 5 = 0
Problem 9 :
Let C be the circle with center at (1, 1) and radius = 1. If T is the circle centered at (0, y) passing through the origin and touching the circle C externally, then the radius of T is equal to
a) √3/√2 b) √3/2 c) 1/2 d) 1/4
Problem 10 :
If the coordinates at one end of a diameter of the circle
x 2 + y 2 - 8x - 4y + c = 0
are (11, 2) the coordinates of the other end are
a) (-5, 2) b) (-3, 2) c) (5, -2) d) (-2, 5)
Problem 11 :
The circle passing through (1, -2) and touching the axis of x at (3, 0) passing through the point
a) (-5, 2) b) (2, -5) c) (5, -2) d) (-2, 5)
1) λ = 0 and λ = -40/9, option a
2) -35 < m < 15, option d
3) k = 5/3, option b
4) √10
5) center of the circle is (4, 7), option a
6) x + 2y = 3
7) r = 2√5
8) x 2 + y 2 - 6y - 7 = 0
9) 1/4.
10) (-3, 2).
11) (5, -2)
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Conic section
A conic section is any of the geometric figures that can arise when a plane intersects a cone . (In fact, one usually considers a "two-ended cone," that is, two congruent right circular cones placed tip to tip so that their axes align.) As is clear from their definition, the conic sections are all plane curves , and every conic section can be described in Cartesian coordinates by a polynomial equation of degree two or less.
- 1.1 Nondegenerate conic sections
- 1.2 Degenerate conic sections
- 2 Definitions of conic sections in terms of a focus and a directrix
- 3 Definitions of conic sections in terms of foci
- 4 Definitions of conic sections in terms of Cartesian coordinates
Classification of conic sections
All conic sections fall into the following categories:
Nondegenerate conic sections
- A circle is the conic section formed when the cutting plane is parallel to the base of the cone or equivalently perpendicular to the axis. (This is really just a special case of the ellipse -- see the next bullet point.)
- An ellipse is formed if the cutting plane makes an angle with an axis that is larger than the angle between the slant height and the axis.
By Klaas van Aarsen - Created as a latex tikzpicturePreviously published: Not published before, CC BY-SA 3.0, https://commons.wikimedia.org/w/index.php?curid=25261046
- A parabola is formed when the cutting plane makes an angle with the axis that is equal to the angle between the element of the cone and the axis.
By Klaas van Aarsen - Own work, CC BY-SA 3.0, https://commons.wikimedia.org/w/index.php?curid=25261094
- An hyperbola is formed when the cutting plane makes an angle with the axis that is smaller than the angle between the element of the cone and the axis.
By Klaas van Aarsen - Own work, CC BY-SA 3.0, https://commons.wikimedia.org/w/index.php?curid=25261095
Degenerate conic sections
If the cutting plane passes through the vertex of the cone, the result is a degenerate conic section. Degenerate conics fall into three categories:
- If the cutting plane makes an angle with the axis that is larger than the angle between the element of the cone and the axis then the plane intersects the cone only in the vertex, i.e. the resulting section is a single point . This is a degenerate ellipse.
- If the cutting plane makes an angle with the axis equal to the angle between the element of the cone and the axis then the plane is tangent to the cone and the resulting section is a line . This is a degenerate parabola.
- If the cutting plane makes an angle with the axis that is smaller than then angle between the element of the cone and the axis then the resulting section is two intersecting lines. This is a degenerate hyperbola.
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There are alternate (but equivalent) definitions of every conic section. We present them here:
Definitions of conic sections in terms of a focus and a directrix
Definitions of conic sections in terms of foci
- Circle - The locus of all points equidistant from a fixed point (i.e. the two foci are the same point).
- Ellipse - The locus of all points where the sum of the distances to two points (the foci) is the same.
- Parabola - The locus of all points where the sum or difference of the distances to a point (the focus) and an infinitely far point (which can be replaced by a line) is the same.
- Hyperbola - The locus of all points where the absolute difference of the distances to two points (the foci) is the same.
Definitions of conic sections in terms of Cartesian coordinates
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A conic section is any of the geometric figures that can arise when a plane intersects a cone. (In fact, one usually considers a "two-ended cone," that is, two congruent right circular cones placed tip to tip so that their axes align.) As is clear from their definition, the conic sections are all plane curves, and every conic section can be ...
This topic covers the four conic sections and their equations: Circle, Ellipse, Parabola, and Hyperbola. ... Solving basic equations & inequalities (one variable, linear) Unit 3. Linear equations, functions, & graphs ... Challenging conic section problems (IIT JEE) Learn. Representing a line tangent to a hyperbola
By definition, a conic section is a curve obtained by intersecting a cone with a plane. In Algebra II, we work with four main types of conic sections: circles, parabolas, ellipses and hyperbolas. Each of these conic sections has different characteristics and formulas that help us solve various types of problems.
If the plane is perpendicular to the axis of revolution, the conic section is a circle. If the plane intersects one nappe at an angle to the axis (other than 90°), then the conic section is an ellipse. Figure 11.5.2: The four conic sections. Each conic is determined by the angle the plane makes with the axis of the cone.
Conic sections: FAQ. Level up on the above skills and collect up to 240 Mastery points. Level up on all the skills in this unit and collect up to 900 Mastery points! When we slice a cone, the cross-sections can look like a circle, ellipse, parabola, or a hyperbola. These are called conic sections, and they can be used to model the behavior of ...
When working with circle conic sections, we can derive the equation of a circle by using coordinates and the distance formula. The equation of a circle is (x - h) 2 + (y - k) 2 = r 2 where r is equal to the radius, and the coordinates (x,y) are equal to the circle center. The variables h and k represent horizontal or vertical shifts in the ...
Unit test. Level up on all the skills in this unit and collect up to 800 Mastery points! The cross-sections of a cone form several interesting curved shapes—circles, ellipses, parabolas, and hyperbolas. Use the distance formula to relate the geometric features of the figures to their algebraic equations.
Problem 1. Identify the conic section represented by the equation \displaystyle 2x^ {2}+2y^ {2}-4x-8y=40 2x2 +2y2 −4x−8y = 40. Then graph the equation. Ellipse. Parabola. Hyperbola. Circle. Problem 2. Identify the conic section represented by the equation.
Conic sections are generated by the intersection of a plane with a cone (Figure 5.5.2 ). If the plane is parallel to the axis of revolution (the y -axis), then the conic section is a hyperbola. If the plane is parallel to the generating line, the conic section is a parabola.
A circle is a set of points (x, y) which are a fixed distance r, the radius, away from a fixed point (h, k), the center. The equation of the circle in standard form is. (x - h) 2 + (y - k) 2 = r 2. If a circle is given in general form then we must complete the square on the x and y parts of the equation to rewrite it in standard form.
Problem Solving in Circle Conic Sections. Intellectual Math. Learn Math step-by-step. PROBLEM SOLVING IN CIRCLE CONIC SECTIONS. Problem 1 : The center of the circle inscribed in a square formed by the lines x 2 - 8x + 12 = 0 and y 2 - 14y + 45 = 0 is. ... So, the radius of the circle T is 1/4. Problem 6 :
Now, conic sections is a situation and are a set of problems that can oftentimes be a bit difficult because there's a lot to remember with conic sections. But in this video and over the course of the next few videos, we're going to be going over some examples and graphs and situations that will hopefully make the concept of conic sections super ...
If you section the cone with a plane that is parallel to the outer surface of the cone the cut edge will be a parabola and if you tilt the cutting plane past that point and on to vertical you will get a hyperbola. So the 'conic sections' are literally the shapes you get when you section a cone.
In this video lesson, we'll be discussing about solving situational problems involving conic sections. Let's begin by CIRCLE and PARABOLA with each problem r...
PRACTICE PROBLEMS ON CIRCLES IN CONIC SECTIONS. Problem 1 : The equation of the circle passing through (1, 5) and (4, 1) and touching y-axis is. x 2 + y 2 - 5x - 6y + 9 + λ(4x+3y-19) = 0. ... Solving Two Step Inequality Word Problems. Read More. Exponential Function Context and Data Modeling.
The conic sections as originally conceived in ancient Greece were "slices" of two cones tip-to-tip. Slicing this figure in different ways produces each of the four conic sections - Circle, Ellipse, Parabola and Hyperbola. This chapter will examine the Circle and the Parabola. 5.1: The Equation of the Circle ...
Step-by-Step Examples. Algebra. Conic Sections. Find the Circle Through (2,5) with Center (3,4) , Step 1. Find the radius for the circle. The radius is any line segment from the center of the circle to any point on its circumference. In this case, is the distance between and .
Conic section from expanded equation: circle & parabola. Sal manipulates the equation x^2+y^2-3x+4y=4 in order to find that it represents a circle, and the equation 2x^2+y+12x+16=0 in order to find it represents a parabola. Created by Sal Khan.
Algebra Examples. This is the form of a circle. Use this form to determine the center and radius of the circle. (x−h)2 +(y−k)2 = r2 ( x - h) 2 + ( y - k) 2 = r 2. Match the values in this circle to those of the standard form. The variable r r represents the radius of the circle, h h represents the x-offset from the origin, and k k ...
A conic section is any of the geometric figures that can arise when a plane intersects a cone. (In fact, one usually considers a "two-ended cone," that is, two congruent right circular cones placed tip to tip so that their axes align.) As is clear from their definition, the conic sections are all plane curves, and every conic section can be described in Cartesian coordinates by a polynomial ...
WAYS TO SOLVE PROBLEMS INVOLVING CONIC SECTIONS | SHS - PRE CALCULUS | JUDD HERNANDEZParabola and circles.Do you like this video? If you like it, you can hel...
‼️FIRST QUARTER‼️🟣 GRADE 11: PROBLEM SOLVING INVOLVING CIRCLES‼️SHS MATHEMATICS PLAYLISTS‼️General MathematicsFirst Quarter: https://tinyurl.com ...