Equation of Circle with Origin as Center

Q. A chord of negative slope is drawn from the point $P(\sqrt{264},0)$ to $x^2+4y^2=16.$ If the chord intersects the curve at $A$ and $B,$ where $O$ is the origin, then which of the following is (are) CORRECT ?

1. The maximum area of $△AOB$ is $4$ sq. units
2. The maximum area of $△AOB$ is $8$ sq. units
3. When the area of $△AOB$ is maximum, then slope of line $AB$ is $\frac{-1}{2\sqrt{2}}$
4. When the area of $△AOB$ is maximum, then slope of line $AB$ is $\frac{-1}{8\sqrt{2}}$

Q.

The equation of a circle which cuts the three circles

$x^2+y^2-3x-6y+14=0,$

$x^2+y^2-x-4y+8=0$

$x^2+y^2+2x-6y+9=0$

orthogonally is $\underset{–}{}$

1. x² + y² - 2x - 4y + 1 = 0

2. x² + y² + 2x + 4y + 1 = 0

3. x² + y² - 2x + 4y + 1 = 0

4. x² + y² - 2x - 4y - 1 = 0

Q.

If one of the diameters of the circle $x^2+y^2–2x–6y+6=0$ is a chord of another circle ‘$C$’ whose center is at $\left(2,1\right)$, then its radius is

Q. Let the abscissae of the two points $P\text{and}Q$ on a circle be the roots of $x^2-4x-6=0$ and the ordinates of $P\text{and}Q$ be the roots of $y^2+2y-7=0$. If $PQ$ is a diameter of the circle $x^2+y^2+2ax+2by+c=0$, then the value of $(a+b-c)$ is

1. $13$
2. $12$
3. $14$
4. $16$

Q.

Find the equation of director circle of the circle whose diameters are 2x - 3y + 12 = 0 and x + 4y - 5 = 0 and area is 154 square units.

1. 

2. 

3. 

4. 

Q.

The equation of the circle passing through the origin which cuts off intercept of length 6 and 8 from the axes is

1. $x^2+y^2-12x-16y=0$

2. $x^2+y^2+12x+16y=0$

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

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

Q. A straight line $L_1:\frac{x}{a}+\frac{y}{b}=1$ intersects the $x$-axis and $y$-axis at $P$ and $Q$ respectively and a straight line $L_2$ perpendicular to $L_1$ cuts the $x$-axis and $y$-axis at $R$ and $S$ respectively. The locus of the point of intersection of the lines $PS$ and $QR$ is

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

Q. The equation of that diameter of the circle $x^2+y^2-6x+2y-8=0$ which passes through the origin is

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

Q. The plane through the intersection of the planes $x+2y+z-1=0$ and $2x+y+3z-2=0$ is perpendicular to the planes $x+y+z-1=0$ and $x+ky+3z-1=0$. Then the value of $k$ is

1. $-\frac{5}{2}$
2. $-\frac{3}{2}$
3. $\frac{5}{2}$
4. $\frac{3}{2}$

Q.

The area in the first quadrant between $x^2+y^2={\mathrm{\pi }}^2$ and$y=\sin x$ is

1. $\frac{{\mathrm{\pi }}^3-8}{4}$

2. $\frac{{\mathrm{\pi }}^3}{4}$

3. $\frac{{\mathrm{\pi }}^3-16}{4}$

4. $\frac{{\mathrm{\pi }}^3-8}{2}$

Q. The equation of the circle having centre at the origin and passing through the point $(3,4),$ is

1. $x^2+y^2=9$
2. $x^2+y^2=16$
3. $x^2+y^2=25$
4. $x^2+y^2+6x+8y+25=0$

Q. The equation of the circle concentric with the circle $x^2+y^2+8x+10y-7=0$ and passing through the centre of the circle $x^2+y^2-4x-6y=0$ is

1. 
2. 
3. 
4. 

Q.

Find the equation of the circle which passes through the origin and cuts off chords of lengths 4 and 6 on the positive side of the x-axis and y-axis respectively.

Q. Consider a region $M$ which contains all the points $(x,y)$ such that $x^2+y^2\le 100$ and $\sin (x+y)\ge 0$ where $x,y\in R$. If the area of region $M$ is $m\pi$ sq units, then the value of $m$ is

Q. In what direction a line be drawn through the point (1, 2) so that its point of intersection with the line x + y = 4 will be at a Distance of $\frac{\sqrt{6}}{3}$

1. 
2. 
3. 
4. 

Q. Let the latus ractum of the parabola $y^2=4x$ be the common chord to the circles $C_1$ and $C_2$ each of them having radius $2\sqrt{5}$. Then, the distance between the centres of the circles $C_1$ and $C_2$ is:

1. $8$
2. $8\sqrt{5}$
3. $4\sqrt{5}$
4. $12$

Q. The transformed equation of $xy=c^2$ when the axis are rotated through an angle $\frac{\pi }{4}$ (in the anti clockwise direction), is

1. $X^2-Y^2=c^2$
2. $X^2-Y^2=2c^2$
3. $X^2+Y^2=c^2$
4. $X^2+Y^2=2c^2$

Q. The number of points belonging to the set $A=\left\{\right(2,-4),(5,-4),(4,-5),(1,-3),(2,-11),(2,-5\left)\right\}$ lying inside the circle $x^2+y^2-4x+12y+15=0$ and below the line $2x-3y=16,$ is

Q. Consider a $△PQR$ in a circle $x^2+y^2=16$ such that $Q\equiv (2\sqrt{2},2\sqrt{2})$ and $R\equiv (-2,2\sqrt{3}).$ Then the measure of $\angle QPR$ is

1. ${75}^{\circ }$
2. ${45}^{\circ }$
3. ${37.5}^{\circ }$
4. ${90}^{\circ }$

Q. A straight line $L_1:\frac{x}{a}+\frac{y}{b}=1$ intersects the $x$-axis and $y$-axis at $P$ and $Q$ respectively and a straight line $L_2$ perpendicular to $L_1$ cuts the $x$-axis and $y$-axis at $R$ and $S$ respectively. The locus of the point of intersection of the lines $PS$ and $QR$ is

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

Q. If $x+y=\cos \theta ,x-y=\sin \theta \left(\theta \text{is parameter}\right)$, then the locus of $P(x,y)$ is

1. a straight line of slope $\frac{1}{2}$
2. a circle with centre $(0,0)$
3. a straight line passing through $(0,0)$
4. a circle with radius $\frac{1}{\sqrt{2}}\text{units}$

Q. For all values of $\theta$, the locus of the point of intersection of the lines $xcos\theta +ysin\theta =a$ and $xsin\theta -ycos\theta =b$ is

1. An ellipse
2. A circle
3. A parabola
4. A hyperbola

Q. The centre of the circle passing through (0, 0) and (1, 0) and touching the circle $x^2+y^2=9$ is

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

Q. Area of the region in which point $p(x,y),\{x>0\}$ , lies; such that $y\le \sqrt{16-x^2}$ and $\left|ta{n}^{-1}\left(\frac{y}{x}\right)\right|\le \frac{\pi }{3}$ is

1. $(4\sqrt{3}-\pi )$
2. $(\sqrt{3}-\pi )$
3. $\left(\frac{16}{3}\pi \right)$
4. $(\frac{8\pi }{3}+8\sqrt{3})$

Q. The triangle $PQR$ of area $A$ is inscribed in the parabola $y^2=4ax$ such that the vertex $P$ lies at the vertex of the parabola and the base $QR$ is a focal chord. The modulus of the difference of the ordinates of the points $Q$ and $R$ is

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

Q. If the coordinates of a point are a tan $(\theta +\alpha )$ and b tan $(\theta +\beta )$, where $\theta$ is a variable, then locus of the point is

1. A hyperbola
2. A rectangular hyperbola
3. An ellipse
4. None of these

Q. Let AB be a chord of the circle $x^2+y^2=r^2$ subtending a right angle at the center. Then, the locus of the centroid of triangle PAB as P moves on the circle is

1. a parabola
2. a circle
3. an ellipse
4. a pair of straight lines

Q. Find by integration, the are of the ellipse $16x^2+25y^2=400$.

Q. Find the area of the region {$(x,y):y^2\le 6ax$ and $x^2+y^2\le 16a^2$} using method of integration.

Q. The area of the region bounded by the curve y $=\frac{16-x^2}{4}$ and $y=se{c}^{-1}[-si{n}^{2}x],$ where [.] stands for the greatest integer function is:

1. $(4-\pi {)}^{3/2}$
2. $\frac{4}{3}(4-\pi {)}^{3/2}$
3. $\frac{8}{3}(4-\pi {)}^{3/2}$
4. $\frac{8}{3}(4-\pi )$