Circle · Mathematics · AP EAPCET

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MCQ (Single Correct Answer)

1
The perimeter of the locus of the point $P$ which divides the line segment QA internally in the ratio $1: 2$, where $A=(4,4)$ and $Q$ lies on the circle $x^2+y^2=9$, is
AP EAPCET 2024 - 21th May Evening Shift
2
If the equation of the circle whose radius is 3 units and which touches internally the circle $x^2+y^2-4 x-6 y-12=0$ at the point $(-1,-1)$ is $x^2+y^2+p x+q y+r=0$, then $p+q-r=$
AP EAPCET 2024 - 21th May Evening Shift
3
The equation of the circle touching the circle $x^2+y^2-6 x+6 y+17=0$ externally and to which the lines $x^2-3 x y-3 x+9 y=0$ are normal is
AP EAPCET 2024 - 21th May Evening Shift
4
The pole of the straight line $9 x+y-28=0$ with respect to the circle $2 x^2+2 y^2-3 x+5 y-7=0$ is
AP EAPCET 2024 - 21th May Evening Shift
5
The equation of a circle which touches the straight lines $x+y=2, x-y=2$ and also touches the circle $x^2+y^2=1$ is
AP EAPCET 2024 - 21th May Evening Shift
6
The radical axis of the circle $x^2+y^2+2 g x+2 f y+c=0$ and $2 x^2+2 y^2+3 x+8 y+2 c=0$ touches the circle $x^2+y^2+2 x+2 y+1=0$. Then,
AP EAPCET 2024 - 21th May Evening Shift
7
$2 x-3 y+1=0$ and $4 x-5 y-1=0$ are the equations of two diameters of the circle $S \equiv x^2+y^2+2 g x+2 f y-11=0 . Q$ and $R$ are the points of contact of the tangents drawn from the point $P(-2,-2)$ to this circle. If $C$ is the centre of the circle $S=0$, then the area (in square units ) of the quadrilateral $P Q C R$ is
AP EAPCET 2024 - 21th May Morning Shift
8
If the inverse point of the point $(-1,1)$ with respect to the circle $x^2+y^2-2 x+2 y-1=0$ is $(p, q)$, then $p^2+q^2=$
AP EAPCET 2024 - 21th May Morning Shift
9
If $(a, b)$ is the mid-point of the chord $2 x-y+3=0$ of the circle $x^2+y^2+6 x-4 y+4=0$, then $2 a+3 b=$
AP EAPCET 2024 - 21th May Morning Shift
10
If a direct common tangent drawn to the circle $x^2+y^2-6 x+4 y+9=0$ and $x^2+y^2+2 x-2 y+1=0$ touches the circles at $A$ and $B$, then $A B=$
AP EAPCET 2024 - 21th May Morning Shift
11

The radius of the circle which cuts the circles $x^2+y^2-4 x-4 y+7=0, x^2+y^2+4 x-4 y+6=0$ and $x^2+y^2+4 x+4 y+5=0$ orthogonally is

AP EAPCET 2024 - 21th May Morning Shift
12
$A(2,3), B(-1,1)$ are two points. If $P$ is a variable point such that $\angle A P B=90^{\circ}$, then locus of $P$ is
AP EAPCET 2024 - 20th May Evening Shift
13
The largest among the distances from the point $P(15,9)$ to the points on the circle $x^2+y^2-6 x-8 y-11=0$ is
AP EAPCET 2024 - 20th May Evening Shift
14
The circle $x^2+y^2-8 x-12 y+\alpha=0$ lies in the first quadrant without touching the coordinate axes. If $(6,6)$ is an interior point to the circle, then
AP EAPCET 2024 - 20th May Evening Shift
15
The equation of the circle whose diameter is the common chord of the circles $x^2+y^2-6 x-7=0$ and $x^2+y^2-10 x+16=0$ is
AP EAPCET 2024 - 20th May Evening Shift
16
If the locus of the mid-point of the chords of the circle $x^2+y^2=25$, which subtend a right angle at the origin is given by $\frac{x^2}{\alpha^2}+\frac{y^2}{\alpha^2}=1$, then $|\alpha|=$
AP EAPCET 2024 - 20th May Evening Shift
17
The radical centre of the circles $x^2+y^2+2 x+3 y+1=0$, $x^2+y^2+x-y+3=0, x^2+y^2-3 x+2 y+5=0$
AP EAPCET 2024 - 20th May Evening Shift
18
If a circle is inscribed in an equilateral triangle of side $a$, then the area of any square (in sq units) inscribed in this circle is
AP EAPCET 2024 - 20th May Morning Shift
19
If the line segment joining the points $(1,0)$ and $(0,1)$ subtends an angle of $45^{\circ}$ at a variable point $P$, then the equation of the locus of $P$ is
AP EAPCET 2024 - 20th May Morning Shift
20

Equation of the circle having its centre on the line $2 x+y+3=0$ and having the lines $3 x+4 y-18=0,3 x+4 y+2=0$ as tangents is

AP EAPCET 2024 - 20th May Morning Shift
21
If power of a point $(4,2)$ with respect to the circle $x^2+y^2-2 \alpha x+6 y+\alpha^2-16=0$ is 9 , then the sum of the lengths of all possible intercepts made by such circles on the coordinate axes is
AP EAPCET 2024 - 20th May Morning Shift
22
Let $\alpha$ be an integer multiple of 8 . If $S$ is the set of all possible values of $\alpha$ such that the line $6 x+8 y+\alpha=0$ intersects the circle $x^2+y^2-4 x-6 y+9=0$ at two distinct points, then the number of elements in $S$ is
AP EAPCET 2024 - 20th May Morning Shift
23
If the circle $x^2+y^2-8 x-8 y+28=0$ and $x^2+y^2-8 x-6 y+25-\alpha^2=0$ have only one common tangent, then $\alpha=$
AP EAPCET 2024 - 20th May Morning Shift
24
If the equation of the circle passing through the points of intersection of the circles $x^2-2 x+y^2-4 y-4=0$, $x^2+2 x+y^2+4 y-4=0$ and the point $(3,3)$ is given by $x^2+y^2+\alpha x+\beta y+\gamma=0$, then $3(\alpha+\beta+\gamma)=$
AP EAPCET 2024 - 20th May Morning Shift
25
The angle subtended by the chord $x+y-1=0$ of the circle $x^2+y^2-2 x+4 y+4=0$ at the origin is
AP EAPCET 2024 - 19th May Evening Shift
26
Let $P$ be any point on the circle $x^2+y^2=25$. Let $L$ be the chord of contact of $P$ with respect to the circle $x^2+y^2=9$. The locus of the poles of the lines $L$ with respect to the circle $x^2+y^2=36$ is
AP EAPCET 2024 - 19th May Evening Shift
27
If the circles $S \equiv x^2+y^2-14 x+6 y+33=0$ and $S^1 \equiv x^2+y^2-a^2=0(a \in N)$ have 4 common tangents, then possible number of values of $a$ is
AP EAPCET 2024 - 19th May Evening Shift
28
If the area of the circum-circle of triangle formed by the line $2 x+5 y+\alpha=0$ and the positive coordinate axes is $\frac{29 \pi}{4} S q$, units, then $|\alpha|=$
AP EAPCET 2024 - 19th May Evening Shift
29
The circle $S \equiv x^2+y^2-2 x-4 y+1=0$ cuts the $Y$-axis at $A, B(O A>O B)$. If the radical axis of $S \equiv 0$ and $S' \equiv x^2+y^2-4 x-2 y+4=0$ cuts the $Y$-axis at $C$, then the ratio in which $C$ divides $A B$ is
AP EAPCET 2024 - 19th May Evening Shift
30
If the circle $S=0$ cuts the circles $x^2+y^2-2 x+6 y=0$, $x^2+y^2-4 x-2 y+6=0$ and $x^2+y^2-12 x+2 y+3=0$ orthogonally, then equation of the tangent at $(0,3)$ on $S=0$ is
AP EAPCET 2024 - 19th May Evening Shift
31
If $\theta$ is the angle between the tangents drawn from the point $(2,3)$ to the circle $x^2+y^2-6 x+4 y+12=0$ then $\theta=$
AP EAPCET 2024 - 18th May Morning Shift
32
If $2 x-3 y+3=0$ and $x+2 y+k=0$ are conjugate lines with respect to the circle $S=x^2+y^2+8 x-6 y-24=0$, then the length of the tangent drawn from the point $\left(\frac{k}{4}, \frac{k}{3}\right)$ to the circle $S=0$, is
AP EAPCET 2024 - 18th May Morning Shift
33
If $Q(h, k)$ is the inverse point of the point $P(1,2)$ with respect to the circle $x^2+y^2-4 x+1=0$, then $2 h+k=$
AP EAPCET 2024 - 18th May Morning Shift
34
If $(a, b)$ and ( $c, d)$ are the internal and external centres of similitudes of the circles $x^2+y^2+4 x-5=0$ and $x^2+y^2-6 y+8=0$ respectively, then $(a+d)(b+q)=$
AP EAPCET 2024 - 18th May Morning Shift
35
A circle $s$ passes through the points of intersection of the circles $x^2+y^2-2 x+2 y-2=0$ and $x^2+y^2+2 x-2 y+1=0$. If the centre of this circle $S$ lies on the line $x-y+6=0$, then the radius of the circle $S$ is
AP EAPCET 2024 - 18th May Morning Shift
36

The locus of mid-points of points of intersection of $$x \cos \theta+y \sin \theta=1$$ with the coordinate axes is

AP EAPCET 2022 - 5th July Morning Shift
37

The radius of the circle having. $$3 x-4 y+4=0$$ and $$6 x-8 y-7=0$$ as its tangents is

AP EAPCET 2022 - 5th July Morning Shift
38

A circle is such that $$(x-2) \cos \theta+(y-2) \sin \theta=1$$ touches it for all values of $$\theta$$. Then, the circle is

AP EAPCET 2022 - 5th July Morning Shift
39

The least distance of the point $$(10,7)$$ from the circle $$x^2+y^2-4 x-2 y-20=0$$ is

AP EAPCET 2022 - 5th July Morning Shift
40

Suppose that the $$x$$-coordinates of the points $$A$$ and $$B$$ satisfy $$x^2+2 x-a^2=0$$ and their $$y$$-coordinates satisfy $$y^2+4 y-b^2=0$$. Then, the equation of the circle with $$A B$$ as its diameter is

AP EAPCET 2022 - 5th July Morning Shift
41

The radical centre of the three circles $$x^2+y^2-1=0, x^2+y^2-8 x+15=0$$ and $$x^2+y^2+10 y+24=0$$ is

AP EAPCET 2022 - 5th July Morning Shift
42

For any real number $$t$$, the point $$\left(\frac{8 t}{1+t^2}, \frac{4\left(1-t^2\right)}{1+t^2}\right)$$ lies on a / an

AP EAPCET 2022 - 4th July Evening Shift
43

The area of the circle passing through the points $$(5, \pm 2),(1,2)$$ is

AP EAPCET 2022 - 4th July Evening Shift
44

The ratio of the largest and shortest distances from the point $$(2,-7)$$ to the circle $$x^2+y^2-14 x-10 y-151=0$$ is

AP EAPCET 2022 - 4th July Evening Shift
45

A circle has its centre in the first quadrant and passes through $$(2,3)$$. If this circle makes intercepts of length 3 and 4 respectively on $$x=2$$ and $$y=3$$, its equation is

AP EAPCET 2022 - 4th July Evening Shift
46

The image of the point $$(3,4)$$ with respect to the radical axis of the circles $$x^2+y^2+8 x+2 y+10=0$$ and $$x^2+y^2+7 x+3 y+10=0$$ is

AP EAPCET 2022 - 4th July Evening Shift
47

The locus of centers of the circles, possessing the same area and having $$3 x-4 y+4=0$$ and $$6 x-8 y-7=0$$ as their common tangent, is

AP EAPCET 2022 - 4th July Morning Shift
48

For any two non-zero real numbers $$a$$ and $$b$$ if this line $$\frac{x}{a}+\frac{y}{b}=1$$ is a tangent to the circle $$x^2+y^2=1$$, then which of the following is true?

AP EAPCET 2022 - 4th July Morning Shift
49

The length of the intercept on the line $$4 x-3 y-10=0$$ by the circle $$x^2+y^2-2 x+4 y-20=0$$ is

AP EAPCET 2022 - 4th July Morning Shift
50

The pole of the line $$\frac{x}{a}+\frac{y}{b}=1$$ with respect to the circle $$x^2+y^2=c^2$$ is

AP EAPCET 2022 - 4th July Morning Shift
51

If the tangent at the point $$P$$ on the circle $$x^2+y^2+6 x+6 y=2$$ meets the straight line $$5 x-2 y+6=0$$ at a point $$Q$$ on the $$Y$$-axis, then the length of $$P Q$$ is

AP EAPCET 2022 - 4th July Morning Shift
52

The locus of a point, which is at a distance of 4 units from $$(3,-2)$$ in $$x y$$-plane is

AP EAPCET 2021 - 20th August Morning Shift
53

Find the equation of the circle which passes through origin and cuts off the intercepts $$-$$2 and 3 over the $$X$$ and $$Y$$-axes respectively.

AP EAPCET 2021 - 20th August Morning Shift
54

The angle between the pair of tangents drawn from $$(1,1)$$ to the circle $$x^2+y^2+4 x+4 y-1=0$$ is

AP EAPCET 2021 - 20th August Morning Shift
55

If the circle $$x^2+y^2-4 x-8 y-5=0$$ intersects the line $$3 x-4 y-m=0$$ in two distinct points, then the number of integral values of '$$m$$' is

AP EAPCET 2021 - 20th August Morning Shift
56

Let $$C$$ be the circle center $$(0,0)$$ and radius 3 units. The equation of the locus of the mid-points of the chords of the circle $$c$$ that subtends an angle of $$\frac{2 \pi}{3}$$ at its centre is

AP EAPCET 2021 - 20th August Morning Shift
57

The length of the common chord of the circles $$x^2+y^2+3x+5y+4=0$$ and $$x^2+y^2+5x+3y+4=0$$ is __________ units.

AP EAPCET 2021 - 20th August Morning Shift
58

Find the equation of the circle which passes through the point $$(1,2)$$ and the points of intersection of the circles $$x^2+y^2-8 x-6 y+21=0$$ and $$x^2+y^2-2 x-15=0$$

AP EAPCET 2021 - 20th August Morning Shift
59

Given, two fixed points $$A(-2,1)$$ and $$B(3,0)$$. Find the locus of a point $$P$$ which moves such that the angle $$\angle A P B$$ is always a right angle.

AP EAPCET 2021 - 19th August Evening Shift
60

The equations of the tangents to the circle $$x^2+y^2=4$$ drawn from the point $$(4,0)$$ are

AP EAPCET 2021 - 19th August Evening Shift
61

If $$P(-9,-1)$$ is a point on the circle $$x^2+y^2+4 x+8 y-38=0$$, then find equation of the tangent drawn at the other end of the diameter drawn through $$P$$

AP EAPCET 2021 - 19th August Evening Shift
62

Find the equation of a circle whose radius is 5 units and passes through two points on the $$X$$-axis, which are at a distance of 4 units from the origin

AP EAPCET 2021 - 19th August Evening Shift
63

If a foot of the normal from the point $$(4,3)$$ to a circle is $$(2,1)$$ and $$2 x-y-2=0$$, is a diameter of the circle, then the equation of circle is

AP EAPCET 2021 - 19th August Evening Shift
64

The length of the tangent from any point on the circle $$(x-3)^2+(y+2)^2=5 r^2$$ to the circle $$(x-3)^2+(y+2)^2=r^2$$ is 16 units, then the area between the two circles in square units is

AP EAPCET 2021 - 19th August Evening Shift
65

The equation of the circle, which cuts orthogonally each of the three circles

$$\begin{aligned} & x^2+y^2-2 x+3 y-7=0, \\ & x^2+y^2+5 x-5 y+9=0 \text { and } \\ & x^2+y^2+7 x-9 y+29=0 \end{aligned}$$

AP EAPCET 2021 - 19th August Evening Shift
66

Find the equations of the tangents drawn to the circle $$x^2+y^2=50$$ at the points where the line $$x+7=0$$ meets it.

AP EAPCET 2021 - 19th August Morning Shift
67

If the chord of contact of tangents from a point on the circle $$x^2+y^2=r_1^2$$ to the circle $$x^2+y^2=r_2^2$$ touches the circle $$x^2+y^2=r_3^2$$, then $$r_1, r_2$$ and $$r_3$$ are in

AP EAPCET 2021 - 19th August Morning Shift
68

Find the equation of the circle passing through $$(1,-2)$$ and touching the $$X$$-axis at $$(3,0)$$.

AP EAPCET 2021 - 19th August Morning Shift
69

Let $$L_1$$ be a straight line passing through the origin and $$L_2$$ be the straight line $$x+y=1$$. If the intercepts made by the circle $$x^2+y^2-x+3 y=0$$ on $$L_1$$ and $$L_2$$ are equal, then which of the following equations represent $$L_1$$

AP EAPCET 2021 - 19th August Morning Shift
70

The radius of the circle whose center lies at $$(1,2)$$ while cutting the circle $$x^2+y^2+4 x+16 y-30=0$$ orthogonally, is units.

AP EAPCET 2021 - 19th August Morning Shift
71

The point which has the same power with respect to each of the circles $$x^2+y^2-8 x+40=0, x^2+y^2-5 x+16=0$$ and $$x^2+y^2-8 x+16 y+160=0$$ is

AP EAPCET 2021 - 19th August Morning Shift
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