If the equation of a circle is 2λ*x*^{2} + (4λ − 6)*y*^{2} − 8*x* + 12*y* − 2 = 0, then the coordinates of centre are _________

Answer :

(2/3, −1)

The equation *x*^{2} + *y*^{2} + 2*x* − 4*y* + 5 = 0 represents *_____________*

Answer :

a point

If the centroid of an equilateral triangle is (1, 1) and its one vertex is (2, 2), then the equation of its circumcircle is *_____________*

Answer :

*x*^{2} + *y*^{2} − 2*x* − 2*y* = 0

The equation of the incircle formed by the coordinate axes and the line 4*x* + 3*y* = 6 is*_____________*

Answer :

4 (*x*^{2} + *y*^{2} − *x* − *y*) + 1 = 0

the circles *x*^{2} + *y*^{2} = 9 and *x*^{2} + *y*^{2} + 8*y* + 2*c* = 0 touch each other, then *c* is equal to *_____________*

Answer :

15

If the circle *x*^{2} + *y*^{2} + 2*ax* + 6*y* + 9 = 0 touches *x*-axis, then the value of *a* is *_____________*

Answer :

± 3

The equation of a circle with radius 4 and touching both the coordinate axes is *_____________*

Answer :

*x*^{2 }+ *y*^{2 }± 8*x* ± 8*y* + 16 = 0

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

Answer :

*x*^{2} + *y*^{2} − 6*x* − 7*y* = 3√2 - 18

The circle *x*^{2} + *y*^{2} + 2*gx* + 2*fy* + *c* = 0 does not intersect *x*-axis, if *g*^{2 }*_____________*

Answer :

< *c *

If (*x*, 3) and (3, 5) are the extremities of a diameter of a circle with centre at (2, *y*), then the values of *x* and *y* are *_____________*

Answer :

*x* = 1, *y* = 4

Equation of the diameter of the circle *x*^{2} + *y*^{2} + 4*x* - 2*y* = 0 which passes through the origin is *_____________*

Answer :

*x* +* 2y* = 0

The vertex of the parabola (*y* + *k*)^{2} = 8*k* (*x* − *k*) is *_____________*

Answer :

(*k*, −*k*)

The equation of the parabola whose vertex is (*k*, 0) and the directrix has the equation *x *+ *y* = 3*k*, is *_____________*

Answer :

*x*^{2} − 2*xy* + *y*^{2} + 6*kx* + 10*ky* – 7*k*^{2} = 0

The locus of the points of trisection of the double ordinates of a parabola is a *_____________*

Answer :

*x*^{2} − 2*xy* + *y*^{2} + 6*kx* + 10*ky* – 7*k*^{2} = 0

The equation of the parabola with focus (0, 0) and directrix *x* + *y* = 7 is *_____________*

Answer :

*x*^{2} + *y*^{2} − 2*xy *+ 14*x* + 14*y* − 49 = 0

In the parabola *y*^{2} = 4*kx*, the length of the chord passing through the vertex and inclined to the axis at π/4 is *_____________*

Answer :

4√2k

The equation 9*x*^{2} + *y*^{2} + 6*xy *− 74*x* − 78*y* + 212 = 0 represents *_____________*

Answer :

a parabola

Which points lie on the parabola *x*^{2} = *ay*?

Answer :

*x* = *at*, *y* = *at*^{2}

Which points lie on the parabola *x*^{2} = 9*ay*?

Answer :

*x* = 3*at*, *y* = 3*at*^{2}

Which points lie on the parabola 16*x*^{2} = 16*ay*?

Answer :

*x* = *at*, *y* = *at*^{2}

Equation of the hyperbola whose vertices are (± 5, 0) and foci at (± 13, 0), is*_____________*

Answer :

144*x*^{2} − 25*y*^{2} = 3600

If *e*_{1} and *e*_{2} are respectively the eccentricities of the ellipse *x*^{2}/18 + *y*^{2}/4 = 1 and the hyperbola *x*^{2}/9 - *y*^{2}/4 = 1, then the relation between *e*_{1} and *e*_{2} is*_____________*

Answer :

2 *e*_{1}^{2} + *e*_{2}^{2} = 3

The distance between the directrices of the hyperbola *x* = 8 sec θ, *y* = 8 tan θ, is*_____________*

Answer :

8√2

The equation of the conic with focus at (1,* −*1) directrix along *x* − *y* + 1 = 0 and eccentricity √2 is*_____________*

Answer :

2*xy* − 4*x* + 4*y* + 1 = 0

The eccentricity of the conic 25*x*^{2} − 144*y*^{2} = 3600 is*_____________*

Answer :

13/12

Answer :

Let (p, q) be the centre of a circle with radius a.

Thus, its equation will be (x−p)^{2} + (y−q)^{2 }= a^{2}

Given:

x^{2 }+ y^{2 }− 4x + 6y = 5

The given equation can be rewritten as follows:

(x−2)^{2}+(y+3)^{2}−4−9=5

⇒(x−2)^{2 }+ (y+3)^{2} =18

Thus, the radius = √18=3√2

Answer :

Let (p, q) be the centre of a circle with radius a.

Thus, its equation will be (x−p)^{2} + (y−q)^{2 }= a^{2}

Given:

(x + 5)^{2} + (y + 1)^{2} = 9

Thus, the radius = 3

Answer :

Let (p, q) be the centre of a circle with radius a.

Thus, its equation will be (x−p)^{2} + (y−q)^{2 }= a^{2}

Given:

x^{2} + y^{2}− x + 2y − 3=0

The given equation can be rewritten as follows:

(x− ½ )^{2}+(y+1)^{2}− ¼ −1−3=0

⇒(x− ½ )^{2 }+ (y+1)^{2} = 17/4

Thus, the centre is ( ½ ,−1)

Answer :

Let (p, q) be the centre of a circle with radius a.

Thus, its equation will be (x−p)^{2} + (y−q)^{2 }= a^{2}

Given:

(x − 1)^{2} + y^{2} = 4

Thus, the radius is 2.

Answer :

Let (p, q) be the centre of a circle with radius a.

Thus, its equation will be (x−p)^{2} + (y−q)^{2 }= a^{2}

Given:

x^{2 }+ y^{2 }− 4x + 6y = 5

The given equation can be rewritten as follows:

(x−2)^{2}+(y+3)^{2}−4−9=5

⇒(x−2)^{2 }+ (y+3)^{2} =18

Thus, the centre is (2, −3).

Answer :

Let (p, q) be the centre of a circle with radius a.

Thus, its equation will be (x−p)^{2} + (y−q)^{2 }= a^{2}

Given:

(x + 5)^{2} + (y + 1)^{2} = 9

Here, p = −5, q = −1

Thus, the centre is (-5, −1).

Answer :

Let (p, q) be the centre of a circle with radius a.

Thus, its equation will be (x−p)^{2} + (y−q)^{2 }= a^{2}

Given:

(x − 1)^{2} + y^{2} = 4

Here, p = 1, q = 0 and a = 2

Thus, the centre is (1, 0)

Find the vertex of the following parabolas : 4x^{2} + y = 0

Answer :

Given:

4x^{2} + y = 0

⇒-y/4=x^{2}

On comparing the given equation with x^{2}=−4ay:

4a=1/4

⇒a=1/16

∴ Vertex = (0, 0)

Answer :

Given:

y^{2} = 8x

On comparing the given equation with y^{2}=4ax:

4a=8⇒a=2

∴ Vertex = (0, 0)

Find the radius of each of the following circles : x^{2} + y^{2 }− x + 2y − 3 = 0.

Answer :

Let (p, q) be the centre of a circle with radius a.

Thus, its equation will be (x−p)^{2} + (y−q)^{2 }= a^{2}

Given:

x^{2} + y^{2}− x + 2y − 3=0

The given equation can be rewritten as follows:

(x− ½ )^{2}+(y+1)^{2}− ¼ −1−3=0

⇒(x− ½ )^{2 }+ (y+1)^{2} = 17/4

Thus, the centre is ( ½ ,−1) and and the radius is √17/2.

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