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1. x2+y2-3x+1=0
2. x2+y2-x+5=0

3. x2+y2-8x+6y=8

4. x2+y2-4x+8y=7

General equation of circle ,  Now find the points of intersection of the two circles. At the intersection,              Putting the value of x1 in the equation of any of the two circle to find y1. Now we have 3 points through which the circle is passing,     ,       ,     Now putting these points in general equation we will have 3...
Engineering
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Let ABC be a triangle whose circumcentre is at P.  If the position vectors of A, B, C

and P are

respectively, then the position vector of the orthocentre of this triangle, is :

• Option 1)

• Option 2)

• Option 3)

• Option 4)

https://www.youtube.com/watch?v=y30g2lfwv_c

Engineering
117 Views   |

The number of common tangents to the circles

• Option 1)

1

• Option 2)

2

• Option 3)

3

• Option 4)

4

Common tangents of two circles -

When two circles touch  each other externally, there are three common tangents, two of them are direct.

- wherein

$C_1=(2,3)$

$C_2=(-3,-9)$

$C_1C_2=\sqrt{25+144}$

$r_1=\sqrt{4+9+12}= 5$

$r_2=\sqrt{9+81-26}= 8$

$\therefore r_1+r_2=C_1C_2$

3 Tangents for circle touching externally

Option 1)

1

Option 2)

2

Option 3)

3

Option 4)

4

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Engineering
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The length of a focal chord of the parabola y2 = 4ax at a distance b from the vertex is C. Then

• Option 1)

2a2 = bc

• Option 2)

a3 = b2c

• Option 3)

ac = b2

• Option 4)

b2c = 4a3

Find intersections A(x1, y1), B(x2, y2) of P & L Eliminate y from (1) & (2): m2x2 - 2am x + m2 a2 = 4ax  m2 x2-2a(m + 2) x + m2 a2 = 0                     ..............(3) x1, x2 are the roots, x1 + x2 = 2a (m+2)/m2 ;; x1 x2 = a2         .........(4) Eliminate x from (1) & (2): y = m(y2/4a) - ma my2 - 4a y-4a2 m = 0                        ..........(5) y1, y2 are the roots, y1 + y2 = 4a/m   ...
Engineering
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The radius of the circle passing through the points (1,2) (5,2)and (5,-2) is

• Option 1)

$5\sqrt{2}$

• Option 2)

$2\sqrt{5}$

• Option 3)

$3\sqrt{2}$

• Option 4)

$2\sqrt{2}$

As we learnt in Equation of a circle - - wherein Circle with centre and radius .    On substituting the  value of  (x, y) as (1, 2), we get              ....................(1)  On substituting the  value of  (x, y) as (5, 2), we get         .........................(2) On substituting the  value of  (x, y) as (5, -2), we get        .......................(3) On subtracting  Eq. (1) - (2), we...
Engineering
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The radius of the circle passing through the points (1,2),(5,2) and (5,-2) is

• Option 1)

$5\sqrt{2}$

• Option 2)

$2\sqrt{5}$

• Option 3)

$3\sqrt{2}$

• Option 4)

$2\sqrt{2}$

As we learnt in

Equation of a circle -

$\left ( x-h \right )^{2}+\left ( y-k \right )^{2}= r^{2}$

- wherein

Circle with centre $\left ( h,k \right )$ and radius $r$.

On substituting the  value of  (x, y) as (1, 2), we get

$(1-h)^{2}+(2-k)^{2}=r^{2}$            ....................(1)

On substituting the  value of  (x, y) as (5, 2), we get

$(5-h)^{2}+(2-k)^{2}=r^{2}$        .........................(2)

On substituting the  value of  (x, y) as (5, -2), we get

$(5-h)^{2}+(-2-k)^{2}=r^{2}$       .......................(3)

On subtracting  Eq. (1) - (2), we get

h = 3

On subtacting Eq. (2) - Eq. (3), we get

k = 0

$\therefore r = 2\sqrt{2}$

On

Option 1)

$5\sqrt{2}$

Incorrect

Option 2)

$2\sqrt{5}$

Incorrect

Option 3)

$3\sqrt{2}$

Incorrect

Option 4)

$2\sqrt{2}$

Correct

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Engineering
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The equation of a tangent to the circle $x^{2}+y^{2}=25\;passes \:through (-2,11)\:is$

• Option 1)

$4x+3y=25$

• Option 2)

$7x-24y=320$

• Option 3)

$3x+4y=38$

• Option 4)

$24x+7y+125=0$

Equation of tangent -   - wherein Tangent to circle   at       is the tangent Now,  Also  On solving ) Option 1) Correct option Option 2) Incorrect Option Option 3) Incorrect Option Option 4) Incorrect Option
Engineering
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The equation of a circle which passes through the point (1,-2) and (4,-3) and whose centre lies on the line 3x+4y=7 is

• Option 1)

$15(x^{2}+y^{2})-84x+18y-55=0$

• Option 2)

$15(x^{2}+y^{2})-94x+18y+55=0$

• Option 3)

$15(x^{2}+y^{2})+84x-18y+55=0$

• Option 4)

None of these

Family of circle - - wherein Equation of the family of circles passing through point of intersection .    Equation of family of circles  center  his on    Option 1)      Incorrect Option 2) Correct Option 3) Incorrect Option 4) None of these Incorrect
Engineering
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The equation of a circle passing through the origin and cutting of intercepts each equal to +5 of the axis is

• Option 1)

$x^{2}+y^{2}+5x-5y=0$

• Option 2)

$x^{2}+y^{2}-5x+5y=0$

• Option 3)

$x^{2}+y^{2}-5x-5y=0$

• Option 4)

$x^{2}+y^{2}+5x+5y=0$

General form of a circle -   - wherein centre = radius =       as it passes trough origin Also  &    so,  Option 1) Incorrect Option 2) Incorrect Option 3) Correct Option 4) Incorrect
Engineering
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The coordinates of the foot of perpendicular from (a,o) on the line $y= mx+\frac{a}{m}$   are

• Option 1)

$\left ( 0,\frac{-a}{m} \right )$

• Option 2)

$\left ( \frac{a}{m},0 \right )$

• Option 3)

$\left ( 0,\frac{a}{m} \right )\:$

• Option 4)

None

General form of the equation of a line - In  , scope of a line = . - wherein   are the constants.    -------------------------(i) step    Step of    so   ------------------------ (ii) Solutions (i) and (ii) Option 1) Incorrect Option 2) Incorrect Option 3) Correct Option 4) None Incorrect
Engineering
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The circles $x^{2}+y^{2}+2x-2y+1=0\:and\:x^{2}+y^{2}-2x-2y+1=0\:touch\:each\:other$

• Option 1)

extrernally at (0,1)

• Option 2)

Internally at (0,1)

• Option 3)

Externally at (1,0)

• Option 4)

Internally at (1,0)

Common tangents of two circles - When two circles touch  each other externally, there are three common tangents, two of them are direct.   - wherein     Option 1) extrernally at (0,1) This is correct option Option 2) Internally at (0,1) This is incorrect option Option 3) Externally at (1,0) This is incorrect option Option 4) Internally at (1,0) This is incorrect option
Engineering
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The circle $x^{2}+y^{2}+4x-7y+12=0$ cuts an intercept on y-axis is equal to

• Option 1)

7

• Option 2)

+4

• Option 3)

+3

• Option 4)

+1

General form of a circle -   - wherein centre = radius =     y-axis intercept X=0   Option 1) 7 This option is incorrect Option 2) +4 This option is incorrect Option 3) +3 This option is incorrect Option 4) +1 This option is correct
Engineering
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The cartesion equation of the curve $x=7+4\:\cos \alpha ,y=-3+4\sin \alpha\:is$

• Option 1)

$x^{2}+y^{2}-14x+6y+42=0$

• Option 2)

$x^{2}+y^{2}-6x+14y+21=0$

• Option 3)

$x^{2}+y^{2}-10x+12y+28=0$

• Option 4)

None of these

Locus - Path followed by a point p(x,y) under given condition (s). - wherein It is satisfied by all the points (x,y) on the locus.     So + Option 1) This is correct option Option 2) This is incorrect option Option 3) This is incorrect option Option 4) None of these  This is incorrect option
Engineering
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The angle between the tangents drawn to the parabola y2 = 12x from the point ( -3, 2 )

• Option 1)

$90^{\circ}$

• Option 2)

$60^{\circ}$

• Option 3)

$30^{\circ}$

• Option 4)

$45^{\circ}$

Option 1) This is correct option Option 2) This is incorrect option Option 3) This is incorrect option Option 4) This is incorrect option
Engineering
111 Views   |

If y=2x+k is a diameter to the circle $2(x^{2}+y^{2})+3x+4y-1=0,then\: K\: equals$

• Option 1)

0

• Option 2)

1

• Option 3)

+2

• Option 4)

$+\frac{1}{2}$

General form of a circle -   - wherein centre = radius =    Centre = so,     Option 1) 0 This is incorrect option Option 2) 1 This is incorrect option Option 3) +2 This is incorrect option Option 4) This is correct option
Engineering
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If y=$2x$ is a chord of the circle $x^{2}+y^{2}=10x,then$ the equation of the circle whose diameter is this chord is

• Option 1)

$x^{2}+y^{2}+2x+4y=0$

• Option 2)

$x^{2}+y^{2}+2x-4y=0$

• Option 3)

$x^{2}+y^{2}-2x-4y=0$

• Option 4)

None of these

Family of circle - - wherein Equation of the family of circles passing through point of intersection .     is a family of circle Centre  it lies on    Option 1) This solution is incorrect Option 2) This solution is incorrect Option 3) This solution is correct Option 4) None of these  This solution is incorrect
Engineering
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If the straight line $y=mx$ is outside the circle

$x^{2}+y^{2}-20y+90=0, then$

• Option 1)

$m< 3$

• Option 2)

$|m|<3$

• Option 3)

$m> 3$

• Option 4)

$|m|>3$

As we learnt in  General form of a circle -   - wherein centre = radius =    distance of (0, 10) from  & radius       Option 1) This solution is incorrect Option 2) This solution is correct Option 3) This solution is incorrect Option 4) This solution is incorrect
Engineering
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If orthocentre and circumcentre  of a $\bigtriangleup$ are respectively (1,1) and (3,2) , then the coordinates of its  centroid  are

• Option 1)

$\left ( \frac{7}{5},\frac{5}{3} \right )$

• Option 2)

$\left ( \frac{5}{3},\frac{7}{3} \right )$

• Option 3)

$\left ( 7,5 \right )$

• Option 4)

None

Euler line - In any non equilateral triangle the circumcentre (O), the centriod (G) and the orthocentre (H) are collinear and G divides OH in ratio 1:2 - wherein    Cr = Centroid; Orthocentre Circumcentre                                                                                                                                          Option 1) This solution is correct Option...
Engineering
86 Views   |

If latus rectum of the hyperbola is half of its transverse axis, then its eccentricity is

• Option 1)

3/2

• Option 2)

$\frac{\sqrt{3}}{2}$

• Option 3)

$\sqrt{\frac{3}{2}}$

• Option 4)

none of these

Also,  Option 1) 3/2 This solution is incorrect Option 2) This solution is incorrect Option 3) This solution is incorrect Option 4) none of these This solution is correct
Engineering
168 Views   |

If focus of a parabola is (2, 0) and one extremity of latus return is (2,2), then its equation is

• Option 1)

y2= 4(3-x)

• Option 2)

y2 = 4x - 4

• Option 3)

both (a) and (b)

• Option 4)

none of these

Standard equation of parabola - - wherein     Option 1) y2= 4(3-x) This solution is incorrect Option 2) y2 = 4x - 4 This solution is incorrect Option 3) both (a) and (b) This solution is correct Option 4) none of these This solution is incorrect
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