Parametric Equation of a Circle

Q.

Let$A(1,4)$ and$B(1,-5)$be two points. Let $P$ be a point on the circle${(x-1)}^2+{(y-1)}^2=1$such that ${\left(PA\right)}^2+{\left(PB\right)}^2$ have maximum value, then the points $P,A\text{and}B$ lie on.

1. a parabola

2. a straight line

3. a hyperbola

4. an ellipse

Q.

Parametric equation of $(x-h{)}^2+(y-k{)}^2=r^{2}arex+h=rcos\theta andy+k=rsin\theta$

1. True

2. False

Q.

Find the parametric equation of the circle $x^2+y^2-2x+4y-11=0$

1. $x=-2+4cos\theta andy=+4+4cos\theta$

2. $x=-2+4cos\theta andy=+4+4cos\theta$

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

4. $x=1+4cos\theta andy=-2+4sin\theta$

Q. What will be the tangent value of acute angle $\theta$ between the two asymptotes of the hyperbola
$\frac{x^2}{49}-\frac{y^2}{25}=1$

1. $\tan \theta =\frac{35}{12}$
2. $\tan \theta =\frac{12}{35}$
3. $\tan \theta =\frac{12}{37}$
4. $\tan \theta =\frac{35}{12}$

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. Let $x$, $y$ be real variables satisfying the equation $x^2+y^2+8x-10y+40=0$. If $a=max\left\{\right(x+2{)}^2+(y-3{)}^2\}$ and $b=min\left\{\right(x+2{)}^2+(y-3{)}^2\}$, then

1. $a+b=18$
2. $a+b=\sqrt{2}$
3. $a-b=4\sqrt{2}$
4. $ab=73$

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 locus of the curve whose perametric equations are $x=a\left(\frac{1-t^2}{1+t^2}\right)$ and $y=\frac{2at}{1+t^2}$, where $t$ being a parameter is

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

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

Q. The locus of the curve whose perametric equations are $x=a\left(\frac{1-t^2}{1+t^2}\right)$ and $y=\frac{2at}{1+t^2}$, where $t$ being a parameter is

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. 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. The area of an equilateral triangle inscribed in the circle $x^2+y^2+2gx+2fy+c=0$ is

1. $\frac{3\sqrt{3}}{2}(g^2+f^2-c)$
2. $\frac{3\sqrt{3}}{4}(g^2+f^2-c)$
3. $\frac{3\sqrt{3}}{4}(g^2+f^2+c)$
4. $\frac{3\sqrt{3}}{2}(g^2+f^2+c)$

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

Q.

Parametric equation of $(x-h{)}^2+(y-k{)}^2=r^{2}arex+h=rcos\theta andy+k=rsin\theta$

1. True

2. False

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 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 $(0,0)$