Equation of a Chord Joining Two Points with Circle in Parametric Form

Q. Let $r_1$ and $r_2$ be the radii of the largest and smallest circles, respectively, which pass through the point $(–4,1)$ and having their centres on the circumference of the circle $x^2+y^2+2x+4y-4=0.$ If
$\frac{r_1}{r_2}=a+b\sqrt{2,}$ then $a+b$ is equal to:

1. $3$
2. $7$
3. $11$
4. $5$

Q. The locus of midpoint of chord of the circle $x^2+y^2-2x-2y-2=0$, which makes an angle of ${120}^{\circ }$ at the centre, is

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

Q. Let $S$ be the circle in the $xy$-plane defined by the equation $x^2+y^2=4$.

Let $E_1{E}_2$ and $F_1{F}_2$ be the chords of $S$ passing through the point $P_0(1,1)$ and parallel to the $x$-axis and the $y$-axis, respectively. Let $G_1{G}_2$ be the chord of $S$ passing through $P_0$ and having slope $-1$. Let the tangents to $S$ at $E_1$ and $E_2$ meet at $E_3$, the tangents to $S$ at $F_1$ and $F_2$ meet at $F_3$, and the tangents to $S$ at $G_1$ and $G_2$ meet at $G_3$. Then, the points $E_3,F_3$, and $G_3$ lie on the curve

1. $(x-4)(y-4)=4$
2. $x+y=4$
3. $(x-4{)}^2+(y-4{)}^2=16$
4. $xy=4$

Q. Three concentric circles of which biggest is $x^2+y^2=1$, have their radii in A.P. If the line $y=x+1$ cuts all the three circles in real and distinct points, then the interval in which the common difference of $AP$ will lie, is

1. $(0,\frac{\sqrt{2}-1}{2\sqrt{2}})$
2. $(0,\frac{\sqrt{2}+1}{2\sqrt{2}})$
3. $(0,\frac{\sqrt{2}-1}{\sqrt{2}})$
4. $(0,\frac{\sqrt{2}-1}{2})$

Q.

If $\left|z\right|=2$, then the points representing the complex numbers $-1+5z$ will lie on a

1. circle

2. straight line

3. parabola

4. None of these

Q. Let $S$ be the circle in the $xy$-plane defined by the equation $x^2+y^2=4$.

Let $E_1{E}_2$ and $F_1{F}_2$ be the chords of $S$ passing through the point $P_0(1,1)$ and parallel to the $x$-axis and the $y$-axis, respectively. Let $G_1{G}_2$ be the chord of $S$ passing through $P_0$ and having slope $-1$. Let the tangents to $S$ at $E_1$ and $E_2$ meet at $E_3$, the tangents to $S$ at $F_1$ and $F_2$ meet at $F_3$, and the tangents to $S$ at $G_1$ and $G_2$ meet at $G_3$. Then, the points $E_3,F_3$, and $G_3$ lie on the curve

1. $x+y=4$
2. $(x-4{)}^2+(y-4{)}^2=16$
3. $(x-4)(y-4)=4$
4. $xy=4$

Q.

Equation of chord AB of circle $x^2+y^2=2$ passing through P(2, 2) such that $\frac{PB}{PA}=3$, is given by

1. $x=3y$

2. $x=y$

3. $y-2=\sqrt{3}(x-2)$

4. $noneofthese$

Q. Let the equation of two concentric circles is $x^2+y^2-2x+4y+\lambda =0$. If a chord of first circle is tangent to second circle and normal to a circle passing through centre of these circles and having a centre at $(2,-1)$ is also touching first circle. Then length of chord with respect to first circle is:

1. $\sqrt{6}$
2. $2\sqrt{6}$
3. $3\sqrt{6}$
4. $4\sqrt{6}$

Q. Let the equation of two concentric circles is $x^2+y^2-2x+4y+\lambda =0$. If a chord of first circle is tangent to second circle and normal to a circle passing through centre of these circles and having a centre at $(2,-1)$ is also touching first circle. Then length of chord with respect to first circle is:

1. $\sqrt{6}$
2. $2\sqrt{6}$
3. $3\sqrt{6}$
4. $4\sqrt{6}$

Q. The length of the chord $AB$ of the circle $x^2+y^2-6x+8y-13=0$ whose midpoint is $(2,-3)$ is (units)

Q. Three concentric circles of which the biggest is $x^2+y^2=1$, have their radii in A.P. If the line y = x + 1 cuts all the circles in real and distinct points. The interval in which the common difference of the A.P. will lie is

1. 
2. 
3. 
4. 

Q. Let $S$ be the circle in the $xy$-plane defined by the equation $x^2+y^2=4$.

Let $E_1{E}_2$ and $F_1{F}_2$ be the chords of $S$ passing through the point $P_0(1,1)$ and parallel to the $x$-axis and the $y$-axis, respectively. Let $G_1{G}_2$ be the chord of $S$ passing through $P_0$ and having slope $-1$. Let the tangents to $S$ at $E_1$ and $E_2$ meet at $E_3$, the tangents to $S$ at $F_1$ and $F_2$ meet at $F_3$, and the tangents to $S$ at $G_1$ and $G_2$ meet at $G_3$. Then, the points $E_3,F_3$, and $G_3$ lie on the curve

1. $x+y=4$
2. $(x-4{)}^2+(y-4{)}^2=16$
3. $(x-4)(y-4)=4$
4. $xy=4$

Q. If $\overrightarrow{a},\overrightarrow{b},\overrightarrow{c}$ are unit vectors, then the maximum value of ${|2\overrightarrow{a}-3\overrightarrow{b}|}^2+{|2\overrightarrow{b}-3\overrightarrow{c}|}^2+{|2\overrightarrow{c}-3\overrightarrow{a}|}^2$ is

Q. Let the orthocentre and centroid of a triangle be $A(-3,5)$ and $B(3,3)$ respectively. If $C$ is the circumcentre of this triangle, then the radius of the circle having line segment $AC$ as diameter, is

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

Q. The centres of a set of circles, each of radius $3$, lie on the circle $x^2+y^2=25$. The locus of any point with such circle is

1. $4\le x^2+y^2\le 64$
2. $x^2+y^2\le 5$
3. $x^2+y^2\le 25$
4. $3\le x^2+y^2\le 9$

Q. The length of the normal chord at (8, 7) on $y^2-2y-4x-3=0$ is

1. $\frac{3\sqrt{10}}{7}$
2. $\frac{45\sqrt{10}}{9}$
3. $\frac{37\sqrt{5}}{11}$
4. $\frac{40\sqrt{10}}{9}$

Q. The vector $3\hat{i}+5\hat{j}+2\hat{k},2\hat{i}-3\hat{j}-5\hat{k}$ and $5\hat{i}+2\hat{j}-3\hat{k}$ form the sides of a/an

1. None of these
2. Equilateral triangle
3. Isoscales triangle
4. Right angled triangle

Q. Three concentric circles of which biggest is $x^2+y^2=1$, have their radii in A.P. If the line $y=x+1$ cuts all the three circles in real and distinct points, then the interval in which the common difference of $AP$ will lie, is

1. $(0,\frac{\sqrt{2}-1}{2\sqrt{2}})$
2. $(0,\frac{\sqrt{2}+1}{2\sqrt{2}})$
3. $(0,\frac{\sqrt{2}-1}{\sqrt{2}})$
4. $(0,\frac{\sqrt{2}-1}{2})$

Q. Find all non-negative integers a, b, c, d, n such that $a^2+b^2+c^2+d^2=7\cdot 4^n$.

1. $(2^{n+1},2^n,2^n,2^n,2^n),(3\cdot 2^n,3\cdot 2^n,3\cdot 2^n)$
2. $(3^{n-1},2^n,2^n,2^n,2^n),(3\cdot 2^n,3\cdot 2^n,3\cdot 2^n)$
3. $(3^{n+1},2^n,2^n,2^n,2^n),(3\cdot 2^n,3\cdot 2^n,3\cdot 2^n)$
4. $(2^{n-1},2^n,2^n,2^n,2^n),(3\cdot 2^n,3\cdot 2^n,3\cdot 2^n)$

Q. For the circle $x^2+y^2+4x-7y+12=0$ the following statement is true

1. the length of tangent from $(1,2)$ is $\sqrt{7}$
2. Intercept on y-axis is $\sqrt{2}$
3. intercept of x-axis is $2+\sqrt{2}$
4. $Noneofthese$

Q. If a chord, which is not a tangent, of the parabola $y^2=16x$ has the equation $2x+y=p$, and midpoint $(h,k)$, then which of the following is(are) of $p,h$ and $k$?

1. $p=5,h=4,k=-3$
2. $p=-1,h=1,k=-3$
3. $p=-2,h=2,k=-4$
4. $p=2,h=3,k=-4$

Q. Let the equation of two concentric circles is $x^2+y^2-2x+4y+\lambda =0$. If a chord of first circle is tangent to second circle and normal to a circle passing through centre of these circles and having a centre at $(2,-1)$ is also touching first circle. Then length of chord with respect to first circle is:

1. $\sqrt{6}$
2. $2\sqrt{6}$
3. $3\sqrt{6}$
4. $4\sqrt{6}$

Q. If the locus of the mid-point of the line segment from the point $(3,2)$ to a point on the circle , $x^2+y^2=1$ is a circle of the radius $r$ then $r$ is equal to :

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

Q.

If the equations of two circles whose radii are a, a' are S= 0 and S'=0, then show that the circles S/a + S'/a' =0 and S/a - S'/a' = 0 intersect orthogonally.