Mathematics

Coordinate Geometry Circles

61 Questions

Coordinate geometry circles cover equations of circles, loci, chords, and intersection points. They are an important part of the mathematics syllabus for advanced tests. Practicing these improves accuracy in algebraic manipulations and coordinate plotting.

circle equationsdiameter and centerlocus of pointsauxiliary circles

Coordinate Geometry Circles Questions

Multiple choice

What is the equation of the circle that passes through the points (2, 3), (4, 5), and (6, 7)?

  1. (x - 4)^2 + (y - 5)^2 = 4

  2. (x - 4)^2 + (y - 5)^2 = 8

  3. (x - 4)^2 + (y - 5)^2 = 12

  4. (x - 4)^2 + (y - 5)^2 = 16
Reveal answer Fill a bubble to check yourself
B Correct answer
Explanation

The equation of a circle that passes through three non-collinear points can be found using the following formula: ((x - h)^2 + (y - k)^2 = r^2), where (h, k) is the center of the circle and r is the radius. Using the three given points, we can find the center and radius of the circle and then write the equation of the circle.

Multiple choice

What is the equation of a circle with center (2, 3) and radius 5?

  1. (x - 2)^2 + (y - 3)^2 = 25

  2. (x - 2)^2 + (y - 3)^2 = 5

  3. (x + 2)^2 + (y + 3)^2 = 25

  4. (x + 2)^2 + (y + 3)^2 = 5
Reveal answer Fill a bubble to check yourself
A Correct answer
Explanation

The equation of a circle with center $(h, k)$ and radius $r$ is given by the formula $(x - h)^2 + (y - k)^2 = r^2$. Substituting the given values, we get $(x - 2)^2 + (y - 3)^2 = 5^2 = 25$.

Multiple choice

What is the equation of the circle that passes through the points (1, 2), (3, 4), and (5, 6)?

  1. x^2 + y^2 - 4x - 6y + 12 = 0

  2. x^2 + y^2 - 4x - 6y + 14 = 0

  3. x^2 + y^2 - 4x - 6y + 16 = 0

  4. x^2 + y^2 - 4x - 6y + 18 = 0
Reveal answer Fill a bubble to check yourself
B Correct answer
Explanation

The equation of a circle that passes through three points (x1, y1), (x2, y2), and (x3, y3) is given by the formula (x - x1)(x - x2) + (y - y1)(y - y2) + (x - x1)(x - x3) + (y - y1)(y - y3) + (x - x2)(x - x3) + (y - y2)(y - y3) = 0. Substituting the given points, we get (x - 1)(x - 3) + (y - 2)(y - 4) + (x - 1)(x - 5) + (y - 2)(y - 6) + (x - 3)(x - 5) + (y - 4)(y - 6) = 0. Expanding and simplifying this equation, we get x^2 + y^2 - 4x - 6y + 14 = 0.

Multiple choice

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

  1. \(x^2 + y^2 = r^2\)
  2. \(\frac{x^2}{a^2} + \frac{y^2}{b^2} = 1\)
  3. \(y^2 = 4px\)
  4. \(\frac{x^2}{a^2} - \frac{y^2}{b^2} = 1\)
Reveal answer Fill a bubble to check yourself
A Correct answer
Explanation

The standard form of the equation of a circle is (x^2 + y^2 = r^2), where (r) is the radius of the circle.

Multiple choice

What is the equation of a great circle?

  1. $$x^2 + y^2 + z^2 = R^2$$
  2. $$x^2 + y^2 - z^2 = R^2$$
  3. $$x^2 - y^2 + z^2 = R^2$$
  4. $$x^2 - y^2 - z^2 = R^2$$
Reveal answer Fill a bubble to check yourself
A Correct answer
Explanation

The equation of a great circle is $$x^2 + y^2 + z^2 = R^2$$, where R is the radius of the Earth.

Multiple choice

What is the equation of a small circle?

  1. $$x^2 + y^2 + z^2 = r^2$$
  2. $$x^2 + y^2 - z^2 = r^2$$
  3. $$x^2 - y^2 + z^2 = r^2$$
  4. $$x^2 - y^2 - z^2 = r^2$$
Reveal answer Fill a bubble to check yourself
A Correct answer
Explanation

The equation of a small circle is $$x^2 + y^2 + z^2 = r^2$$, where r is the radius of the small circle.

Multiple choice

Find the equation of the circle with center ((2, -3)) and radius (5).

  1. \((x - 2)^2 + (y + 3)^2 = 25\)
  2. \((x - 2)^2 + (y + 3)^2 = 15\)
  3. \((x - 2)^2 + (y + 3)^2 = 20\)
  4. \((x - 2)^2 + (y + 3)^2 = 30\)
Reveal answer Fill a bubble to check yourself
A Correct answer
Explanation

The equation of a circle with center ((h, k)) and radius (r) is: ((x - h)^2 + (y - k)^2 = r^2). Plugging in the values of (h = 2, k = -3, r = 5), we get: ((x - 2)^2 + (y + 3)^2 = 5^2 = 25).

Multiple choice

Which of the following equations represents a circle?

  1. x^2 + y^2 = 9

  2. x^2 - y^2 = 9

  3. x^2 + y^2 - 9 = 0

  4. x^2 - y^2 - 9 = 0
Reveal answer Fill a bubble to check yourself
A Correct answer
Explanation

The standard equation of a circle with center at the origin and radius r is given by x^2 + y^2 = r^2. Therefore, the equation x^2 + y^2 = 9 represents a circle with center at the origin and radius 3.

Multiple choice

Which of the following is the equation of a tangent line to the circle x^2 + y^2 = 25 at the point (3, 4)?

  1. y = 4x - 7

  2. y = -4x + 7

  3. y = 3x - 4

  4. y = -3x + 4
Reveal answer Fill a bubble to check yourself
B Correct answer
Explanation

The slope of the tangent line to a circle at a given point is equal to the negative reciprocal of the slope of the radius vector from the center of the circle to that point. The radius vector from the center (0, 0) to the point (3, 4) has a slope of 4/3. Therefore, the slope of the tangent line is -3/4. Substituting this slope and the point (3, 4) into the point-slope form of a line, we get: y - 4 = (-3/4)(x - 3), which simplifies to y = -4x + 7.

Multiple choice

Which of the following is the equation of the sphere with center at (2, -1, 3) and radius 4?

  1. (x - 2)^2 + (y + 1)^2 + (z - 3)^2 = 16

  2. (x + 2)^2 + (y - 1)^2 + (z + 3)^2 = 16

  3. (x - 2)^2 + (y + 1)^2 + (z - 3)^2 = 4

  4. (x + 2)^2 + (y - 1)^2 + (z + 3)^2 = 4
Reveal answer Fill a bubble to check yourself
A Correct answer
Explanation

The standard equation of a sphere with center at (h, k, l) and radius r is given by (x - h)^2 + (y - k)^2 + (z - l)^2 = r^2. Substituting the values of the center and radius, we get: (x - 2)^2 + (y + 1)^2 + (z - 3)^2 = 4^2 = 16. Therefore, the equation of the sphere with center at (2, -1, 3) and radius 4 is (x - 2)^2 + (y + 1)^2 + (z - 3)^2 = 16.

Multiple choice

Find the equation of the circle with center $(2, -3)$ and radius $5$.

  1. $(x - 2)^2 + (y + 3)^2 = 25$
  2. $(x - 2)^2 + (y + 3)^2 = 16$
  3. $(x - 2)^2 + (y + 3)^2 = 9$
  4. $(x - 2)^2 + (y + 3)^2 = 4$
Reveal answer Fill a bubble to check yourself
A Correct answer
Explanation

The equation of a circle with center $(h, k)$ and radius $r$ is given by the formula $(x - h)^2 + (y - k)^2 = r^2$. Substituting the given values, we get $(x - 2)^2 + (y + 3)^2 = 5^2 = 25$. Therefore, the equation of the circle is $(x - 2)^2 + (y + 3)^2 = 25$.

Multiple choice

What is the equation of a circle with center (h, k) and radius r?

  1. $$(x - h)^2 + (y - k)^2 = r^2$$
  2. $$(x + h)^2 + (y + k)^2 = r^2$$
  3. $$(x - h)^2 - (y - k)^2 = r^2$$
  4. $$(x + h)^2 - (y + k)^2 = r^2$$
Reveal answer Fill a bubble to check yourself
A Correct answer
Explanation

The equation of a circle with center (h, k) and radius r is given by $$(x - h)^2 + (y - k)^2 = r^2$$. This equation represents the set of all points in the plane that are equidistant from the point (h, k).

Multiple choice

What is the equation of a circle with center at the origin and radius r?

  1. $$x^2 + y^2 = r^2$$
  2. $$(x - h)^2 + (y - k)^2 = r^2$$
  3. $$(x + h)^2 + (y + k)^2 = r^2$$
  4. $$(x - h)^2 - (y - k)^2 = r^2$$
Reveal answer Fill a bubble to check yourself
A Correct answer
Explanation

The equation of a circle with center at the origin and radius r is given by $$x^2 + y^2 = r^2$$. This equation represents the set of all points in the plane that are equidistant from the origin.

Multiple choice

What is the equation of a circle that passes through the points (x1, y1) and (x2, y2)?

  1. $$(x - x1)(x - x2) + (y - y1)(y - y2) = 0$$
  2. $$(x - x1)^2 + (y - y1)^2 = (x2 - x1)^2 + (y2 - y1)^2$$
  3. $$(x - x1)^2 - (y - y1)^2 = (x2 - x1)^2 - (y2 - y1)^2$$
  4. $$(x + x1)^2 + (y + y1)^2 = (x2 + x1)^2 + (y2 + y1)^2$$
Reveal answer Fill a bubble to check yourself
B Correct answer
Explanation

The equation of a circle that passes through the points (x1, y1) and (x2, y2) is given by $$(x - x1)^2 + (y - y1)^2 = (x2 - x1)^2 + (y2 - y1)^2$$. This equation represents the set of all points in the plane that are equidistant from the points (x1, y1) and (x2, y2).

Multiple choice

What is the equation of a circle that is tangent to the x-axis at the point (a, 0)?

  1. $$(x - a)^2 + y^2 = a^2$$
  2. $$(x + a)^2 + y^2 = a^2$$
  3. $$(x - a)^2 - y^2 = a^2$$
  4. $$(x + a)^2 - y^2 = a^2$$
Reveal answer Fill a bubble to check yourself
A Correct answer
Explanation

The equation of a circle that is tangent to the x-axis at the point (a, 0) is given by $$(x - a)^2 + y^2 = a^2$$. This equation represents the set of all points in the plane that are equidistant from the point (a, 0).