Parametric Equation of a Circle

Q. The parametric eqauation of the circle $x^2+y^2-2x-4y-4=0$ is .

1. $x=1-3\cos \theta ,y=2+3\sin \theta$
2. $x=1+3\cos \theta ,y=2-3\sin \theta$
3. $x=1+3\cos \theta ,y=2+3\sin \theta$
4. $x=1-3\cos \theta ,y=2-3\sin \theta$

Q.

A line $y=mx+1$ intersects the circle ${(x-3)}^2+{(y+2)}^2=25$ at the points P and Q. If the midpoint of the line segment PQ has x-coordinate $\frac{-3}{5}$, then which one of the following options is correct?

1. $6\le m<8$

2. $2\le m<4$

3. $4\le m<6$

4. -$3\le m<-1$

Q. If $x^2+y^2=25,$ then the maximum value of ${\log }_5|3x+4y|$ is

Q.

If $\omega$ and ${\omega }^2$ are the two imaginary cube roots of unity, then the equation, whose roots are $a{\omega }^{317}$ and $a{\omega }^{382}$, is

1. $x^2+a^{2}x+a^2=0$

2. $x^2+a^{2}x+a=0$

3. $x^2-ax+a^2=0$

4. $x^2-a^{2}x+a=0$

Q. Consider the circle $x^2+y^2=a^2$. Let $A(a,0)$ and $D$ be a given interior point of the circle. If $BC$ be an arbitrary chord of the circle through point $D$, then the locus of the centroid of $\Delta ABC$ is

1. a circle whose radius is less than $\frac{2a}{3}$ units
2. a circle whose radius is greater than $\frac{2a}{3}$ units
3. a circle whose radius is equal to $\frac{2a}{3}$ units
4. None of the above

Q. Consider the family of lines $(x-y-6)+\lambda (2x+y+3)=0$ and $(x+2y-4)+\mu (3x-2y-4)=0.$ If the lines of these two families are at right angle to each other, then the locus of their point of intersection is

1. $x^2+y^2+3x+4y-3=0$
2. $x^2+y^2=25$
3. $x^2+y^2+6x+8y-3=0$
4. $x^2+y^2-3x+4y-3=0$

Q. The standard equation of the circle whose parametric equation are $x=5-5\sin t$ and $y=4+5\cos t$ is

1. $(x+5{)}^2+(y-4{)}^2=25$
2. $(x-5{)}^2+(y-4{)}^2=25$
3. $(x-5{)}^2+(y+4{)}^2=25$
4. $(x+5{)}^2+(y+4{)}^2=25$

Q. If the parametric equation of a circle are $x=-4+5\cos \theta$ and $y=-3+5\sin \theta$, then which of the following is/are true

1. centre of circle is $(4,3)$
2. centre of circle is $(-4,-3)$
3. area of circle is $25\pi$ sq.units
4. perimeter of circle is $10\pi$ units

Q. The locus of the point of intersection of the lines $x=a\left(\frac{1-t^2}{1+t^2}\right)$ and $y=\frac{2at}{1+t^2}$ represents, $t$ being a parameter

1. a circle with a radius of $a$ units
2. a circle passing through origin
3. pair of lines passsing through $(a,a)$
4. pair of lines passsing through $(0,0)$

Q. A circle touches both the circles $x^2+y^2=25$ and $(x-2{)}^2+y^2=1.$ Then locus of its centre is

1. an ellipse with focus $(2,0)$ and $(0,10)$
2. an ellipse with length of major axis $6$ unit
3. an ellipse with eccentricity $\frac{1}{4}$
4. an ellipse with auxiliary circle $x^2+y^2-2x-8=0$

Q.

The center of the circle $x=2+3\cos \theta ,y=3\sin \theta -1$ is

1. $\left(3,3\right)$

2. $\left(2,-1\right)$

3. $\left(-2,1\right)$

4. $\left(1,-2\right)$

Q. Parametric equation of the circle $x^2+y^2-6x+8y+16=0$ are

1. $x=3+3\cos \theta ,y=-4+3\sin \theta$
2. $x=3+3\cos \theta ,y=-4+3\sin \theta$
3. $x=-3+3\cos \theta ,y=-4+3\sin \theta$
4. $x=-3-3\cos \theta ,y=-4-3\sin \theta$

Q. The point $P$ moves in the plane of a regular hexagon such that the sum of the squares of its distances from the vertices of the hexagon is $6a^2$. If the radius of the circumcircle of the hexagon is $r(<a)$, then the locus of $P$ is

1. a circle of radius $a$
2. a circle of radius $\sqrt{a^2+r^2}$
3. a circle of radius $\sqrt{a^2-r^2}$
4. a circle of radius $ar$

Q. Consider the circle $x^2+y^2=a^2$. Let $A(a,0)$ and $D$ be a given interior point of the circle. If $BC$ be an arbitrary chord of the circle through point $D$, then the locus of the centroid of $\Delta ABC$ is

1. a circle whose radius is less than $\frac{2a}{3}$ units
2. a circle whose radius is greater than $\frac{2a}{3}$ units
3. a circle whose radius is equal to $\frac{2a}{3}$ units
4. None of the above

Q. A line has intercepts a, b on axes when the axes are rotated through an angle $\alpha$, the line makes equal intercepts on axes then $\tan \alpha =$

1. 
2. 
3. 
4. 

Q. Equation of the circle having centre at $(3,-1)$ and making an intercept of length $6$ units on the line $2x-5y+18=0$, is

1. $x^2+y^2-6x+2y-38=0$
2. $x^2+y^2-6x+2y-18=0$
3. $x^2+y^2-6x+2y-28=0$
4. $x^2+y^2-6x+2y-48=0$

Q. The parametric eqauation of the circle $x^2+y^2-2x-4y-4=0$ is .

1. $x=1-3\cos \theta ,y=2+3\sin \theta$
2. $x=1+3\cos \theta ,y=2-3\sin \theta$
3. $x=1+3\cos \theta ,y=2+3\sin \theta$
4. $x=1-3\cos \theta ,y=2-3\sin \theta$

Q. The parametric equation of the circle whose center is $(3,-5)$ and touches the $x-$ axis is

1. $x=3+5\cos \theta ,y=-5-5\sin \theta$
2. $x=3+5\cos \theta ,y=-5+5\sin \theta$
3. $x=3+3\cos \theta ,y=-5+3\sin \theta$
4. $x=3+5\cos \theta ,y=-5-5\sin \theta$

Q. The centre of the circle $x=1+2\cos \theta ,y=-3+2\sin \theta ,$ is

1. $(-1,3)$
2. $(1,-3)$
3. $(2,3)$
4. $(-\frac{1}{2},\frac{3}{2})$

Q. The standard equation of the circle whose parametric equation are $x=5-5\sin t$ and $y=4+5\cos t$ is

1. $(x+5{)}^2+(y-4{)}^2=25$
2. $(x-5{)}^2+(y-4{)}^2=25$
3. $(x-5{)}^2+(y+4{)}^2=25$
4. $(x+5{)}^2+(y+4{)}^2=25$

Q. The parametric coordinates of the circle whose center coordinates are $(-4,3)$ and touches the $y$-axis, is

1. $x=-4+4\cos \theta ,y=3+4\sin \theta$
2. $x=-4+4\cos \theta ,y=-3+4\sin \theta$
3. $x=4-4\cos \theta ,y=3+4\sin \theta$
4. $x=4+4\cos \theta ,y=3+4\sin \theta$

Q. Equation of circle whose parametric equations are $x=5-5\sin t$ and $y=4+5\cos t$ is

1. $(x+5{)}^2+(y-4{)}^2=25$
2. $(x-5{)}^2+(y+4{)}^2=25$
3. $(x+5{)}^2+(y+4{)}^2=25$
4. $(x-5{)}^2+(y-4{)}^2=25$

Q. The parametric form of the circle $x^2+y^2-4(x+y)=8$ is

1. $x=2+4\cos \theta ,y=2+4\sin \theta$
2. $x=2-2\cos \theta ,y=2-2\sin \theta$
3. $x=-2+2\cos \theta ,y=-2+2\sin \theta$
4. $x=-2-4\cos \theta ,y=-2-4\sin \theta$

Q. Consider a circle with parametric equation $x=\frac{a}{\sqrt{2}}+a\cos \theta ,$ $y=\frac{a}{\sqrt{2}}+a\sin \theta .$ Which of the following is/are TRUE ?

1. circle passes through the origin
2. radius of the circle is $\frac{a}{\sqrt{2}}$ units
3. circle passes through $(\frac{a}{2},\frac{a}{2})$
4. radius of the circle is $a$ units

Q. A circle whose parametric coordinates are given by $x=\frac{a}{\sqrt{2}}+a\cos \theta$ and $y=\frac{a}{\sqrt{2}}+a\sin \theta$, then choose correct option(s) among the following.

1. will pass through origin
2. will have the radius $\frac{a}{\sqrt{2}}$ units
3. will pass through $(\frac{a}{\sqrt{2}},\frac{a}{\sqrt{2}})$
4. will have the radius of $a$ units

Q. The distance of the point $(2,3)$ from the line $2x-3y+9=0$ measured along a line $x-y+1=0$ is

1. $4\sqrt{2}$
2. $2\sqrt{2}$
3. $\frac{1}{\sqrt{2}}$
4. $\sqrt{2}$

Q. If a straight line through $C(-\sqrt{8},\sqrt{8})$ making an angle ${135}^{\circ }$ with the x-axis cuts the circle $x=5cos\theta ,y=5sin\theta$ in points A and B, then length of segment AB is

1. 5
2. 10
3. 15
4. $15\sqrt{2}$

Q. Two curves $a_i{x}^2+b_i{y}^2=1;i=1,2$ where $a_1\ne a_2,b_1\ne b_2,a_1,a_2,b_1,b_2\ne 0$ may intersect orthogonally if _____

1. $a_1{a}_2=b_1{b}_2$
2. $a_1^{-1}+a_2^{-1}=b_1^{-1}+b_2^{-1}$
3. $a_1^{-1}-a_2^{-1}=b_1^{-1}-b_2^{-1}$
4. $a_1{b}_2=a_2{b}_1$

Q. A line $y=mx+1$ intersects the circle $(x-3{)}^2+(y+2{)}^2=25$ at points $P$ and $Q$.If the mid point of line segment $PQ$ has $x$- coordinate equal to $\frac{-3}{5}$, then which of the following options is correct

1. $6\le m<8$
2. $4\le m<6$
3. $2\le m<4$
4. $-3\le m<-1$

Q. Parametric equation of the circle $x^2+y^2-6x+8y+16=0$ are

1. $x=3+3\cos \theta ,y=-4+3\sin \theta$
2. $x=3+3\cos \theta ,y=-4+3\sin \theta$
3. $x=-3+3\cos \theta ,y=-4+3\sin \theta$
4. $x=-3-3\cos \theta ,y=-4-3\sin \theta$