Find the equation of the circle which touches the both axes in first quadrant and whose radius is $a$.

Show that the point $(x, y)$ given by $x=\frac{2 a t}{1+t^2}$ and $y=\frac{a\left(1-t^2\right)}{1+t^2}$ lies on a circle.

If a circle passes through the points $(0,0),(a, 0)$ and $(0, b)$, then find the coordinates of its centre.

Find the equation of the circle which touches $X$-axis and whose centre is $(1,2)$.

If the lines $3 x+4 y+4=0$ and $6 x-8 y-7=0$ are tangents to a circle, then find the radius of the circle.

Find the equation of a circle which touches both the axes and the line $3 x-4 y+8=0$ and lies in the third quadrant.

If one end of a diameter of the circle $x^2+y^2-4 x-6 y+11=0$ is $(3,4)$, then find the coordinates of the other end of the diameter.

Find the equation of the circle having $(1,-2)$ as its centre and passing through $3 x+y=14,2 x+5 y=18$.

If the line $y=\sqrt{3} x+k$ touches the circle $x^2+y^2=16$, then find the value of $k$.

Find the equation of a circle concentric with the circle $x^2+y^2-6 x+12 y+15=0$ and has double of its area.

If the latusrectum of an ellipse is equal to half of minor axis, then find its eccentricity.

If the ellipse with equation $9 x^2+25 y^2=225$, then find the eccentricity and foci.

If the eccentricity of an ellipse is $\frac{5}{8}$ and the distance between its foci is 10, then find latusrectum of the ellipse.

Find the equation of ellipse whose eccentricity is $\frac{2}{3}$, latusrectum is 5 and the centre is $(0,0)$.

Find the distance between the directrices of ellipse $\frac{x^2}{36}+\frac{y^2}{20}=1$.

Find the coordinates of a point on the parabola $y^2=8 x$, whose focal distance is 4 .

17 Find the length of the line segment joining the vertex of the parabola $y^2=4 a x$ and a point on the parabola, where the line segment makes an angle $\theta$ to the $X$-axis.

If the points $(0,4)$ and $(0,2)$ are respectively the vertex and focus of a parabola, then find the equation of the parabola.

If the line $y=m x+1$ is tangent to the parabola $y^2=4 x$, then find the value of $m$.

If the distance between the foci of a hyperbola is 16 and its eccentricity is $\sqrt{2}$, then obtain the equation of the hyperbola.

21 Find the eccentricity of the hyperbola $9 y^2-4 x^2=36$.

Find the equation of the hyperbola with eccentricity $\frac{3}{2}$ and foci at $( \pm 2,0)$.

If the lines $2 x-3 y=5$ and $3 x-4 y=7$ are the diameters of a circle of area 154 square units, then obtain the equation of the circle.

Find the equation of the circle which passes through the points $(2,3)$ and $(4,5)$ and the centre lies on the straight line $y-4 x+3=0$.

25 Find the equation of a circle whose centre is $(3,-1)$ and which cuts off a chord 6 length 6 units on the line $2 x-5 y+18=0$.

26 Find the equation of a circle of radius 5 which is touching another circle $$x^2+y^2-2 x-4 y-20=0 \text { at }(5,5) \text {. }$$

Find the equation of a circle passing through the point $(7,3)$ having radius 3 units and whose centre lies on the line $y=x-1$.

Find the equation of each of the following parabolas

(i) directrix $=0$, focus at $(6,0)$

(ii) vertex at $(0,4)$, focus at $(0,2)$

(iii) focus at $(-1,-2)$, directrix $x-2 y+3=0$

Find the equation of the set of all points the sum of whose distances from the points $(3,0),(9,0)$ is 12.

Find the equation of the set of all points whose distance from $(0,4)$ are $\frac{2}{3}$ of their distance from the line $y=9$.

Show that the set of all points such that the difference of their distances from $(4,0)$ and $(-4,0)$ is always equal to 2 represent a hyperbola.

Find the equation of the hyperbola with

(i) Vertices $( \pm 5,0)$, foci $( \pm 7,0)$

(ii) Vertices $(0, \pm 7), e=\frac{7}{3}$.

(iii) Foci $(0, \pm \sqrt{10})$, passing through $(2,3)$.

The line $x+3 y=0$ is a diameter of the circle $x^2+y^2+6 x+2 y=0$.

The shortest distance from the point $(2,-7)$ to the circle $x^2+y^2-14 x-10 y-151=0$ is equal to 5.

If the line $l x+m y=1$ is a tangent to the circle $x^2+y^2=a^2$, then the point $(l, m)$ lies on a circle.

The point (1, 2) lies inside the circle $x^2+y^2-2x+6y+1=0$.

The line $l x+m y+n=0$ will touch the parabola $y^2=4 a x$, if $\mathrm{In}=a m^2$.

If $P$ is a point on the ellipse $\frac{x^2}{16}+\frac{y^2}{25}=1$ whose foci are $S$ and $S^{\prime}$, then $P S+P S^{\prime}=8$.

The line $2 x+3 y=12$ touches the ellipse $\frac{x^2}{9}+\frac{y^2}{4}=2$ at the point $(3,2)$.

The locus of the point of intersection of lines $\sqrt{3} x-y-4 \sqrt{3 k}=0$ and $\sqrt{3} k x+k y-4 \sqrt{3}=0$ for different value of $k$ is a hyperbola whose eccentricity is 2 .

The equation of the circle having centre at $(3,-4)$ and touching the line $5 x+12 y-12=0$ is ............. .

The equation of the circle circumscribing the triangle whose sides are the lines $y=x+2,3 y=4 x, 2 y=3 x$ is ............ .

An ellipse is described by using an endless string which is passed over two pins. If the axes are 6 cm and 4 cm , the length of the string and distance between the pins are .......... .

The equation of the ellipse having foci $(0,1),(0,-1)$ and minor axis of length 1 is $\qquad$ .

The equation of the parabola having focus at $(-1,-2)$ and directrix is $x-2 y+3=0$, is ........... .

The equation of the hyperbola with vertices at $(0, \pm 6)$ and eccentricity $\frac{5}{3}$ is ........... and its foci are ........... .

The area of the circle centred at $(1,2)$ and passing through the point $(4,6)$ is

Equation of a circle which passes through $(3,6)$ and touches the axes is

Equation of the circle with centre on the $Y$-axis and passing through the origin and the point $(2,3)$ is

The equation of a circle with origin as centre and passing through the vertices of an equilateral triangle whose median is of length $3 a$ is

If the focus of a parabola is $(0,-3)$ and its directrix is $y=3$, then its equation is

If the parabola $y^2=4 a x$ passes through the point $(3,2)$, then the length of its latusrectum is

If the vertex of the parabola is the point $(-3,0)$ and the directrix is the line $x+5=0$, then its equation is

If equation of the ellipse whose focus is $(1,-1)$, then directrix the line $x-y-3=0$ and eccentricity $\frac{1}{2}$ is

The length of the latusrectum of the ellipse $3 x^2+y^2=12$ is

If $e$ is eccentricity of the ellipse $\frac{x^2}{a^2}+\frac{y^2}{b^2}=1($ where, $a

The eccentricity of the hyperbola whose latusrectum is 8 and conjugate axis is equal to half of the distance between the foci is

The distance between the foci of a hyperbola is 16 and its eccentricity is $\sqrt{2}$. Its equation is

Equation of the hyperbola with eccentricity $\frac{3}{2}$ and foci at $( \pm 2,0)$ is