Circle Geometry and Equations Quiz

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Questions and Answers

What is the general form of the equation for a circle given the conditions mentioned?

x² + y² - 2gx - 2fy + c = 0

x² + y² + gx + fy = 1

x² + y² + g + f + c = 0

x² + y² + 2gx + 2fy + c = 0 (correct)

Which condition must be true for a general second degree equation in two variables to represent a circle?

a ≠ b and h ≠ 0

a ≠ b and h = 0

a = b and h ≠ 0

a = b and h = 0 (correct)

If the center of a circle lies on the x-axis and has a radius of 13, what point must satisfy the equation of the circle passing through (4, 5)?

(0, 0)

(0, 13) (correct)

(0, 5)

(0, -13)

Which of the following equations represents a circle of radius 13 passing through the point (4, 5)?

x² + y² - 32x - 105 = 0 Signup and view all the answers

What would the center coordinates be for a circle that passes through (7, 3) with a radius of 3 units and whose center lies on the line y = x - 1?

(3, 2) Signup and view all the answers

What is the radius of a circle that touches both axes and passes through the point (1, 1)?

2 Signup and view all the answers

If a circle touches the y-axis at (0, 2) and passes through the point (-1, 0), what is the x-coordinate of another point it passes through?

(-4, 0) Signup and view all the answers

What is the equation of the circle inscribed in a triangle formed by the coordinate axes and the line 3x + 4y = 24?

(x - 8)^2 + (y - 6)^2 = 36 Signup and view all the answers

Which point does not lie on the circle that touches both axes and passes through the point (1, 1)?

(0, 0) Signup and view all the answers

What is the geometric significance of the point (0, 2) in relation to the discussed circles?

A point where it touches the axis Signup and view all the answers

What is the equation of a circle with center at (1, 0) and radius 5?

(x − 1)² + y² = 25 Signup and view all the answers

Which of the following represents the correct transformation of the standard circle equation x² + y² = 1 when the center is moved to (1, 2)?

(x - 1)² + (y - 2)² = 1 Signup and view all the answers

For a circle with the diameter passing through points (0, 0) and (2, 2), what is the center of this circle?

(1, 1) Signup and view all the answers

Which equation correctly represents the circle with center (1, 2) and a diameter of 2?

(x − 1)² + (y − 2)² = 4 Signup and view all the answers

What is true about the normal line of a circle?

It passes through the center of the circle. Signup and view all the answers

What is the center of the circle defined by the equation $x^2 + y^2 - 4x + 4y - 28 = 0$?

(2, -2) Signup and view all the answers

Which of the following is the equation of a circle that is concentric with the circle $x^2 + y^2 - 6x + 12y + 15 = 0$ and has double its area?

$x^2 + y^2 - 3x + 12y - 30 = 0$ Signup and view all the answers

What is the radius of a circle with the equation $x^2 + y^2 - 4x - 2y = eta - 5$ as defined in the problem?

2 Signup and view all the answers

What is the general form of the equation of a circle?

$x^2 + y^2 + 2gx + 2fy + c = 0$ Signup and view all the answers

In the equation $x^2 + y^2 + 2gx + 2fy + c = 0$, what do the variables $g$ and $f$ represent?

Coordinates of the center Signup and view all the answers

What condition describes a circle that touches the X-axis at a point?

The radius is equal to the distance from the center to the X-axis Signup and view all the answers

If a circle has the equation $x^2 + y^2 + 4y - 8 = 0$, what is the radius of the circle?

3 Signup and view all the answers

What is the effect of multiplying the equation of a circle by a positive scalar?

It increases the radius Signup and view all the answers

What is the equation of the circle whose diameter endpoints are (1, 0) and (0, 1)?

x^2 + y^2 - x - y = 0 Signup and view all the answers

Which form of the equation represents a circle with radius r?

(x - x1)^2 + (y - y1)^2 = r^2 Signup and view all the answers

What is the geometric significance of the diametric form of the equation of a circle?

It relates the abscissae to the ordinates of the endpoints. Signup and view all the answers

In the parametric form of a circle, how are the coordinates of any point on the circle defined?

x = r cosθ, y = r sinθ Signup and view all the answers

If A(cos ⍺, sin ⍺), B(cos β, sin β), and C(cos γ, sin γ) are vertices of triangle ABC, what does this represent?

The orthocenter coordinates of triangle ABC. Signup and view all the answers

What will be the minimum radius when passing through points (1, 0) and (0, 1)?

1 unit Signup and view all the answers

What type of line can intersect a circle according to the intercepts made by it?

A secant line Signup and view all the answers

In the context of the general equation of a circle, what does the term (x - x1)^2 + (y - y1)^2 represent?

The distance to the center from any point. Signup and view all the answers

Which point does the circle with its center on the line $2x - 3y + 12 = 0$ also pass through?

(-3, 6) Signup and view all the answers

What form does a family of circles passing through the points A(x1, y1) and B(x2, y2) take?

(x - x1)(x - x2) + (y - y1)(y - y2) = 0 Signup and view all the answers

If a circle bisects the circumference of another circle, how can the relationship between their equations be represented?

S + λL = 0 Signup and view all the answers

To find the equation of a family of circles tangent to a given line at a point A(x1, y1), which part of the equation represents the line?

L = 0 Signup and view all the answers

What is the general equation for a circle touching the line $x + y + 1 = 0$ at the point (1, -2)?

$(x - 1)^{2} + (y + 2)^{2} + λ(x + y + 1) = 0$ Signup and view all the answers

In the family of circles passing through (1, 1) and (2, 2), which variable denotes an arbitrary constant?

λ Signup and view all the answers

What is the significance of the variable 'd' in the context of a circle bisecting another circle's circumference?

It is a parameter related to the second circle. Signup and view all the answers

Study Notes

JEE 2024 Circles One Shot

The JEE 2024 Circles One Shot video covers various aspects of circles, including their standard equations, intercepts, notations, and the position of points relative to a circle.
The video also touches on families of circles and chords of circles.
A crash course batch for JEE Main 2024, led by experienced educators, is also advertised.
Standard equations of a circle are presented, including the central form: (x - x₁)² + (y - y₁)² = r². The center is (x₁, y₁) and the radius is r.
A circle centered at (0, 0) with radius r has the equation x² + y² = r².
Examples of finding circle equations given specific centers and radii, or points on the circle, are shown.
The diameter or normal of a circle passes through its center.
Practice problems involving circles, including finding the equation of a circle given certain conditions (radius, center on x or y axis, passing through a point), are shown.

Standard Equations of a Circle

The general equation of a circle is ax² + 2hxy + by² + 2gx + 2fy + c = 0.
A circle is represented by x² + y² + 2gx + 2fy + c = 0, where the center is (-g, -f) and radius is √(g² + f² − c).
Examples of finding the center and radius of a circle given its general equation are shown.

Intercepts Made by a Circle

If a circle intersects a line, AB is the length of the intercept.
The length of an intercept made by a circle on the x or y axis can be determined if the general equation of the circle is given as x² + y² +/- 2gx + 2fy + c = 0. This length is calculated as 2√(g² - c) where g is the term with x and c the constant.
A similar formula exists for intercept length on the y axis.
A circle's center will lie on a line cutting the circle's intercept, if the line is the chord of the circle

Family of Circles

A family of circles includes circles that share common properties. The family can be defined in various ways, for example, it can pass through the intersections of two circles or through the intersection of a line and a circle.

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Description

Test your knowledge on the properties and equations of circles. This quiz covers conditions for circle equations, calculations involving center points, radii, and relationships with coordinate geometry. Perfect for students studying geometry concepts related to circles.