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If in a parallelogram ABDC, the coordinates of A,B and C are respectively (1,2), (3,4) and (2,5) , then the equation of the diagonal AD is:

• Option 1)

$5x-3y+1=0$

• Option 2)

$3x-5y+1=0$

• Option 3)

$5x+3y-11=0$

• Option 4)

$3x+5y-13=0$

Mid-point formula -   - wherein If the point P(x,y) is the mid point of line joining A(x1,y1) and B(x2,y2) .     Two – point form of a straight line -   - wherein The lines passes through    and    As BD and AC are parallel ..............................(1) As AB and CD are parallel ..............................(2) Solving (1) and (2) m=4 and n=7              Option 1)Option 2)Option...

In an ellipse, with centre at the origin, if the difference of the lengths of major axis and minor axis is $10$ and one of the foci is at $\left ( 0,5\sqrt{3} \right )$, then the length of its latus rectum is :

• Option 1)

$10$

• Option 2)

$5$

• Option 3)

$8$

• Option 4)

$6$

focus is at   given difference of major axis-minor axis  Length of LR = Option 1)Option 2)Option 3)Option 4)

If the angle of intersection at a point where two circles with radii $5\: cm$ and $12\: cm$ intersects is $90^{\circ}$, then the length (in cm) of their common chord is :

• Option 1)

$\frac{13}{2}$

• Option 2)

$\frac{13}{5}$

• Option 3)

$\frac{120}{13}$

• Option 4)

$\frac{60}{13}$

Length of common chord =  Option 1)Option 2)Option 3)Option 4)

If the line $x-2y=12$ is the tangent to the ellipse

$\frac{x^{2}}{a^{2}}+\frac{y^{2}}{b^{2}}=1$ at the point $(3,\frac{-9}{2})$ , then the length

of the latus rectum of the ellipse is :

• Option 1)

9

• Option 2)

$12\sqrt2$

• Option 3)

5

• Option 4)

$8\sqrt3$

Tangent to a given ellipse at  Equation of tangent at  Now compare this equation with given equation of tangent  x - 2y = 12 Length of LR =  So, correct  option is (1). Option 1) 9 Option 2) Option 3) 5 Option 4)

If the circles $x^{2}+y^{2}+5Kx+2y+K=0$ and

$2(x^{2}+y^{2})+2Kx+3y-1=0$ , $(K\epsilon R)$ , intersect

at the points P and Q , then the line $4x+5y-K=0$ passes

through P and Q , for :

• Option 1)

infinitely many values of $K$

• Option 2)

no value of $K$

• Option 3)

exactly two values of $K$

• Option 4)

exactly one value of $K$

Given two circles are      Equation of common chord  => ................(1) Given equation of chord is  ..................................(2) On Comparing (1) & (2) There is no value of k  So, option (2) is correct. Option 1) infinitely many values of  Option 2) no value of  Option 3) exactly two values of  Option 4) exactly one value of

If the line $y=mx+7\sqrt{3}$ is normal to the hyperbola $\frac{x^{2}}{24}-\frac{y^{2}}{18}=1$   , then a value of $m$ is :

• Option 1)

$\frac{\sqrt{5}}{2}$

• Option 2)

$\frac{\sqrt{15}}{2}$

• Option 3)

$\frac{2}{\sqrt{5}}$

• Option 4)

$\frac{3}{\sqrt{5}}$

given hyperbola       Normal to hyperbola is slope form            compare this                       Option 1)            Option 2) Option 3) Option 4)

The common tangent to the circle $x^{2}+y^{2}=4\:\:and\:\:x^{2}+y^{2}+6x+8y-24=0$  also passes through the point :

• Option 1)

$(4,-2)$

• Option 2)

$(-6,4)$

• Option 3)

$(6,-2)$

• Option 4)

$(-4,6)$

common tangent will  be   Option 1) Option 2) Option 3) Option 4)

The straight line L at a distance of 4 units from the origin makes

positive intercepts on the coordinate axes and the perpendicular

from the origin to this line makes an angle of $60^{\circ}$ with the line

$x+y=0$ . Then an equation of the line L is :

• Option 1)

$x+\sqrt3y=8$

• Option 2)

$(\sqrt3+1)x+(\sqrt3-1)y=8\sqrt2$

• Option 3)

$\sqrt3x+y=8$

• Option 4)

$(\sqrt3-1)x+(\sqrt3+1)y=8\sqrt2$

Hence, the equation of line is  Option 1) Option 2) Option 3) Option 4)

A triangle has a vertex at (1,2) and the mid points of the two sides

through it are (-1,1) and (2,3). Then the centroid of this triangle is :

• Option 1)

$(1,\frac{7}{3})$

• Option 2)

$(\frac{1}{3},2)$

• Option 3)

$(\frac{1}{3},1)$

• Option 4)

$(\frac{1}{3},\frac{5}{3})$

Centroid of triangle =   Option 1) Option 2) Option 3) Option 4)

The equation of a common tangent to the curves, $y^{2}=16x$ and $xy=-4$, is:

• Option 1)

x-y+4=0

• Option 2)

x+y+4=0

• Option 3)

x-2y+16=0

• Option 4)

2x-y+2=0

Equation of tangent to parabola,  is .............(1) It is tangent to ....................(2) Solving (1) and (2) we will get  For tangent   Putting m = 1 in eqn (1) => equation of common tangent is                                               =>    Option 1) x-y+4=0 Option 2) x+y+4=0 Option 3) x-2y+16=0 Option 4) 2x-y+2=0

A circle touching the x-axis at (3, 0) and making an intercept of length 8 on the y-axis passes through the point:

• Option 1)

(3,10)

• Option 2)

(3,5)

• Option 3)

(2,3)

• Option 4)

(1,5)

=> equation of circle is  Hence (3,10) will satisfy the equation.      Option 1) (3,10) Option 2) (3,5)   Option 3) (2,3)   Option 4) (1,5)

An ellipse, with foci at (0,2) and (0, -2) and minor axis of length 4, passes through which of the following points?

• Option 1)

$(\sqrt{2},2)$

• Option 2)

$(2,\sqrt{2})$

• Option 3)

$(2,2\sqrt{2})$

• Option 4)

$(1,2\sqrt{2})$

let         is the equation of ellipse , focii  Given : 2a = 4 => a = 2  equation of ellipse is   => it passes through  so, option 2 is correct.   Option 1)       Option 2) Option 3) Option 4)

If the normal to the ellipse $3x^{2}+4y^{2}=12$ at a point P on its parallel to the line, $2x+y=4$ and the tangent to the ellipse at P passes through  $Q\left ( 4,4 \right )$ then PQ is equal to :

• Option 1)

$\frac{\sqrt{157}}{2}$

• Option 2)

$\frac{\sqrt{221}}{2}$

• Option 3)

$\frac{\sqrt{61}}{2}$

• Option 4)

$\frac{5\sqrt{5}}{2}$

So,   Let                Equation of normal is  Equation of tangent is it passes through                   Hence point is  Option 1)      Option 2)   Option 3)      Option 4)

Let P be the point of intersection of the common tangents to the parabola $y^{2}=12x$ and the hyperbola $8x^{2}-y^{2}=8$. If S and S' denote the foci of the hyperbola where S lies on the positive x-axis then P divides SS' in a ratio :

• Option 1)

$14:13$

• Option 2)

$5:4$

• Option 3)

$13:11$

• Option 4)

$2:1$

Parabola  hyperbola   Equation of tangents Since they are common tangent                                               Eccentricity of hyperbola So ratio   Option 1) Option 2) Option 3) Option 4)

The locus of the centres of the circles, which touch the circle,

$x^{2}+y^{2}=1$ externally , also touch the y-axis and lie in the

• Option 1)

$x=\sqrt{1+4y},y\geq 0$

• Option 2)

$y=\sqrt{1+2x},x\geq 0$

• Option 3)

$y=\sqrt{1+4x},x\geq 0$

• Option 4)

$x=\sqrt{1+2y},y\geq 0$

Let the centre of one such circle be P(h,k) since it touches y-axis in the first quadrant. => its radius = h Now, since it touches  =>  =>  Hence the required locus is  So, option (2) is correctOption 1)Option 2)Option 3)Option 4)

Lines are drawn parallel  to the line $4x-3y+2=0$, at a

distance $\frac{3}{5}$ from the origin. Then which one of the following points

lies on any of these lines?

• Option 1)

$(\frac{-1}{4},\frac{2}{3})$

• Option 2)

$(\frac{1}{4},\frac{-1}{3})$

• Option 3)

$(\frac{1}{4},\frac{1}{3})$

• Option 4)

$(\frac{-1}{4},\frac{-2}{3})$

Line parallel to 4x-3y+2=0 can be written as, 4x-3y+c=0. This line lies at a distance of   from origin. So, equation of lines are 4x-3y+3=0 and 4x-3y-3=0 Now, putting all options in these equations (1)          (2)    (3)    (4)  Option 1) Option 2) Option 3) Option 4)

If the line $ax+y=c$ , touches both the curves $x^{2}+y^{2}=1$

and $y^{2}=4\sqrt2x$ , then $|c|$ is equal to :

• Option 1)

2

• Option 2)

$\frac{1}{\sqrt2}$

• Option 3)

$\frac{1}{2}$

• Option 4)

$\sqrt2$

line  curve eqns  &  A tangent to the parabola    can be taken as ()    ....................(1)   ()   It will also touch the given circle  if  =>                                    (eqn  of line touching the circle) So, option (4) is correct.   Option 1) 2 Option 2) Option 3) Option 4)

If $5x+9=0$ is the directrix of the hyperbola $16x^{2}-9y^{2}=144,$

then its corresponding focus is :

• Option 1)

(5,0)

• Option 2)

$(\frac{-5}{3},0)$

• Option 3)

$(\frac{5}{3},0)$

• Option 4)

(-5 , 0)

eqn of hyperbola  Directrix of hyperbola  Now, from the eqn of hyperbola  ,  As we know eccentricity =  Required focus is                                                    Option (4) is correct answer.  Option 1)(5,0)Option 2)Option 3)Option 4)(-5 , 0)

If a directrix of a hyperbola centered at the origin and passing

through the point $(4,-2\sqrt3)$ is $5x=4\sqrt5$ and its

eccentricity is e , then :

• Option 1)

$4e^{4}-24e^{2}+27=0$

• Option 2)

$4e^{4}-12e^{2}-27=0$

• Option 3)

$4e^{4}-24e^{2}+35=0$

• Option 4)

$4e^{4}+8e^{2}-35=0$

Std eqn of Hyperbola it passes through  So, => ..........................(1) given eqn of directrix =>  Also, => ...................................(2) And we know that   ................................(3) from (1),(2) and (3) =>  => correct option (3) Option 1) Option 2) Option 3) Option 4)

The line x = y touches a circle at the point (1,1). If the circle

also passes through the point (1, - 3) , then its radius is :

• Option 1)

3

• Option 2)

$2\sqrt2$

• Option 3)

2

• Option 4)

$3\sqrt2$

From tha family of circle and line circle touch the line  at point (1,1) .................(1) It passes through ( 1 , -3 )  Put   in (1)  correct option is (2)      Option 1) 3 Option 2) Option 3) 2 Option 4)
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