Sample QuestionsThe Circle questions
One sample from each question group in this chapter. Select any group above to see the full set with answer keys.
The equation of the incircle formed by the coordinate axes and the line $4x + 3y = 6$ is:
- A
$x^2 + y^2 - 6x - 6y + 9 = 0$
- ✓
$4 (x^2 + y^2 - x - y) + 1 = 0$
- C
$4 (x^2 + y^2 + x + y) + 1 = 0$
- D
Answer: B.
View full solution →If the centroid of an equilateral triangle is $(1, 1)$ and its one vertex is $(-1, 2),$ then the equation of its circumcircle is:
- ✓
$x^2 + y^2 - 2x - 2y - 3 = 0$
- B
$x^2 + y^2 + 2x - 2y - 3 = 0$
- C
$x^2 + y^2 + 2x + 2y - 3 = 0$
- D
Answer: A.
View full solution →The equation of the circle which touches the axes of coordinates and the line $\frac{\text{x}}{3}+\frac{\text{y}}{4}=1$ and whose centres lie in the first quadrant is $x^2 + y^2 − 2cx − 2cy + c^2 = 0,$ where $c$ is equal to:
Answer: D.
View full solution →The equation of the circle passing through the point $(1, 1)$ and having two diameters along the pair of lines $x^2 - y^2 - 2x + 4y - 3 = 0,$ is:
- ✓
$x^2 + y^2 - 2x - 4y + 4 = 0$
- B
$x^2 + y^2 + 2x + 4y - 4 = 0$
- C
$x^2 + y^2 - 2x + 4y + 4 = 0$
- D
Answer: A.
View full solution →If the point $(2, k)$ lies outside the circles $x^2 + y^2 + x - 2y - 14 = 0$ and $x^2 + y^2 = 13$ then klies in the interval:
- A
$(-3,\ -2)\cup(3,\ 4)$
- B
$-3,\ 4$
- ✓
$(-\infty,\ -3)\cup(4,\ \infty)$
- D
$(-\infty,\ -2)\cup(3,\ \infty)$
Answer: C.
View full solution →If the line $y = mx$ does not intersect the circle $(x + 10)^2 + (y + 10)^2 = 180$, then write the set of values taken by $m$.
View full solution →Write the area of the circle passing through (-2, 6) and having its centre at (1, 2).
Write the coordinates of the centre of the circle inscribed in the square formed by the lines x = 2, x = 6, y = 5 and y = 9.
Find the equation of the circle with:
Centre (a, b) and radius $\sqrt{\text{a}^2+\text{b}^2}$
View full solution →If the radius of the circle $x^2 + y^2 + ax + (1 - a) y + 5 = 0$ does not exceed $5$, write the number of integral values a.
View full solution →Find the equation of a circle,
Which touches both the axes and passes through the point $(2, 1)$.
View full solution →Find the equation of the circle passing through the point of intersection of the lines $x + 3y = 0$ and $2x - 7y = 0$ and whose centre is the point of intersection of the lines $x + y + 1 = 0$ and $x - 2y + 4 = 0$.
View full solution →Find the centre and radius of the following circles:
$\text{x}^2+\text{y}^2-4\text{x}+6\text{y}=5$
View full solution →Find the equation of the circle having $(1, −2)$ as its centre and passing through the intersection of the lines $3x + y = 14$ and $2x + 5y = 18$.
View full solution →Find the coordinates of the centre and radius of each of the following circles:$\frac{1}{2}(\text{x}^2+\text{y}^2)+\text{x}\cos\theta+\text{y}\sin\theta-4=0$
View full solution →Find the equation of the circle which touches the axes and whose centre lies on $x - 2y = 3.$
View full solution →Find the equation of the circle passing through the points:
$(5, 7), (8, 1)$ and $(1, 3)$
View full solution →Find the equation of the circle passing through the points:
$(0, 0), (-2, 1)$ and $(-3, 2)$
View full solution →Find the equation of the circle which passes through $(3, -2), (-2, 0)$ and has its centre on the line $2x - y = 3$
View full solution →Find the equation of the circle passing through the points:
$(1, 2), (3, -4)$ and $(5, -6)$
View full solution →Find the equation of the circle which passes through the points $(2, 3)$ and $(4,5)$ and the centre lies on the straight line $y - 4x + 3 = 0$
View full solution →Show that the point $(\text{x},\ \text{y})$ given by $\text{x}=\frac{2\text{at}}{1+\text{t}^2}$ and $\text{y}=\text{a}\Big(\frac{1-\text{t}^2}{1+\text{t}^2}\Big)$2 lies on a circle for all real values of t such that $-1\leq\text{t}\leq1,$ where a is any given real number.
View full solution →Find the equation of the circle which circumscribes the triangle formed by the lines
$y = x + 2, 3y = 4x$ and $2y = 3x$.
View full solution →Prove that the centres of the three circles $x^2+ y^2 - 4x - 6y - 12 = 0 , x^2 + y^2 + 2x + 4y - 10 = 0$ and $x^2 + y^2 - 10x - 16y - 1 = 0$ are collinear.
View full solution →Find the equation of the circle the end points of whose diameter are the centres of the circles $x^2 + y^2 + 6x - 14y - 1 = 0$ and $x^2 + y^2 - 4x + 10y - 2 = 0$.
View full solution →Find the equation of the circle which passes through the points $(2, 3)$ and $(4,5)$ and the centre lies on the straight line $y - 4x + 3 = 0$
View full solution →Show that the point $(\text{x},\ \text{y})$ given by $\text{x}=\frac{2\text{at}}{1+\text{t}^2}$ and $\text{y}=\text{a}\Big(\frac{1-\text{t}^2}{1+\text{t}^2}\Big)$2 lies on a circle for all real values of t such that $-1\leq\text{t}\leq1,$ where a is any given real number.
View full solution →Find the equation of the circle which circumscribes the triangle formed by the lines
$y = x + 2, 3y = 4x$ and $2y = 3x.$
View full solution →Prove that the centres of the three circles $x^2 + y^2 - 4x - 6y - 12 = 0, x^2 + y^2 + 2x + 4y - 10 = 0$ and $x^2 + y^2 - 10x - 16y - 1 = 0$ are collinear.
View full solution →Find the equation of the circle the end points of whose diameter are the centres of the circles $x^2 + y^2 + 6x - 14y - 1 = 0$ and $x^2 + y^2 - 4x + 10y - 2 = 0.$
View full solution →