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12th Maths Weekly Test -4 (Two Dimensional Analytical Geometry-II)

St. Britto Hr. Sec. School - Madurai

12th Maths Weekly Test -4 (Two Dimensional Analytical Geometry-II)-Aug 2020

Time : 1.00 Hours

Part - A

Multiple Choice Question - Answer All Question

The equation of the circle passing through(1,5) and (4,1) and touching y -axis is x²+y²−5x−6y+9+(4x+3y−19)=0 whereλ is equal to

\(0,-\frac { 40 }{ 9 } \)
0
\(\frac { 40 }{ 9 } \)
\(\frac { -40 }{ 9 } \)

The circle x²+y²=4x+8y+5intersects the line3x−4y=m at two distinct points if

15< m < 65
35< m <85
−85<m < −35
−35<m <15

The radius of the circle3x²+by²+4bx−6by+b² =0 is

1
3
\( \sqrt {10}\)
\( \sqrt {11}\)

The equation of the normal to the circle x²+y²−2x 2y+1=0 which is parallel to the line
2x+4y=3 is

x+2y=3
x+2y+3= 0
2x+4y+3=0
x−2y+3= 0

The radius of the circle passing through the point(6,2) two of whose diameter arex+y=6
and x+2y=4 is

10
\( {2} \sqrt {5}\)
6
4

If the normals of the parabola y² = 4x drawn at the end points of its latus rectum are tangents
to the circle (x−3)²+(y+2)²=r² , then the value of r² is

2
3
1
4

The ellipse E₁\(\frac { { x }^{ 2 } }{ 9 } +\frac { { y }^{ 2 } }{ 4 } =1\) is inscribed in a rectangle R whose sides are parallel to the coordinate axes. Another ellipse E₂ passing through the point(0,4) circumscribes the rectangle R . The eccentricity of the ellipse is

\(\frac { \sqrt { 2 } }{ 2 } \)
\(\frac { \sqrt { 3 } }{ 2 } \)
\(\frac { 1 }{ 2 } \)
\(\frac { 3 }{ 4 } \)

The equation of the circle passing through the foci of the ellipse \(\frac { { x }^{ 2 } }{ 16 } +\frac { { y }^{ 2 } }{ 9 } =1\) having centre at
(0,3) is

x²+y²−6y−7=0
x²+y²−6y+7=0
x²+y²−6y−5=0
x²+y²−6y+5=0

Consider an ellipse whose centre is of the origin and its major axis is along x-axis. If its eccentrcity is \(\frac { 3 }{ 5 } \) and the distance between its foci is 6, then the area of the quadrilateral inscribed in the ellipse with diagonals as major and minor axis of the ellipse is

8
32
80
40

An ellipse hasOB as semi minor axes, F and F′ its foci and the angle FBF′ is a right angle. Then the eccentricity of the ellipse is

\(\frac { 1 }{ \sqrt { 2 } } \)
\(\frac { 1 }{ 2 } \)
\(\frac { 1 }{ 4 } \)
\(\frac { 1 }{ \sqrt { 3 } } \)

If the two tangents drawn from a point P to the parabola y² = 4x are at right angles then the
locus of P is

2x+1=0
x = −1
2x−1=0
x =1

The locus of a point whose distance from (-2,0) is \(\frac { 2 }{ 3 } \) times its distance from the line x = \(\frac { -9 }{ 2 } \) is

a parabola
a hyperbola
an ellipse
a circle

If the coordinates at one end of a diameter of the circle x²+y²−8x−4y+c = 0 are (11,2) ,
the coordinates of the other end are

(-5,2)
(2,-5)
(5,-2)
(-2,5)

The equation of the directrix of the parabola y²+ 4y + 4x + 2 = 0 is

x = -1
x = 1
x = \(\frac{-3}{2}\)
x = \(\frac{3}{2}\)

y²- 2x - 2y + 5 = 0 is a

circle
parabola
ellipse
hyperbola

The eccentricity of the ellipse 9x²+ 5y² - 30y= 0 is

\(\frac13\)
\(\frac23\)
\(\frac34\)
none of these

If the distance between the foci is 2 and the distance between the direction is 5, then the equation of the ellipse is

6x² + 10y² = 5
6x²+ 10y² = 15
x² + 3y² = 10
none

The equation 7x²- 6\(\sqrt { 3 } \) xy + 13y² - 4\(\sqrt { 3 } \) x - 4y - 12 = 0 represents

parabola
ellipse
hyperbola
rectangular hyperbola

If the foci of the ellipse \(\frac { { x }^{ 2 } }{ 16 } +\frac { { y }^{ 2 } }{ { b }^{ 2 } } \) = 1 and the hyperbola \(\frac { { x }^{ 2 } }{ 144 } -\frac { { y }^{ 2 } }{ 81 } =\frac { 1 }{ 25 } \) coincide then b² is

1
5
7
9

The director circle of the ellipse \(\frac { { x }^{ 2 } }{ 9 } +\frac { { y }^{ 2 } }{ 5 } =1\) is

x² + y² = 4
x² +y² = 9
x² +y² = 45
x² +y² = 14

The length of the diameter of a circle with centre (1, 2) and passing through (5, 5) is

5
\(\sqrt{45}\)
10
\(\sqrt{50}\)

The area of the circle (x - 2)² + (y - k)² = 25 is

25ㅠ
5ㅠ
10ㅠ
25

If y = 2x + c is a tangent to the circle x² + y² = 5, then c is

土5
土\(\sqrt5\)
土5\(\sqrt2\)
±2\(\sqrt5\)

The angle between the tangents drawn from (1, 4) to the parabola y² = 4x is __________

\(\frac { \pi }{ 2 } \)
\(\frac { \pi }{ 3 } \)
\(\frac { \pi }{ 5 } \)
\(\frac { \pi }{ 5 } \)

If e₁,e₂ are eccentricities of the ellipse \(\frac { { x }^{ 2 } }{ { a }^{ 2 } } -\frac { { y }^{ 2 } }{ { b }^{ 2 } } \) = 1 and the hyperbola \(\frac { { x }^{ 2 } }{ { a }^{ 2 } } -\frac { { y }^{ 2 } }{ { b }^{ 2 } } \) = 1 then

\({ e }_{ 1 }^{ 2 }\) - \({ e }_{ 2 }^{ 2 }\) = 1
\({ e }_{ 1 }^{ 2 }\) + \({ e }_{ 2 }^{ 2 }\) = 1
\({ e }_{ 1 }^{ 2 }\) - \({ e }_{ 2 }^{ 2 }\) = 2
\({ e }_{ 1 }^{ 2 }\) - \({ e }_{ 2 }^{ 2 }\)=2

The number of normals to the hyperbola \(\frac { { x }^{ 2 } }{ { a }^{ 2 } } -\frac { { y }^{ 2 } }{ { b }^{ 2 } } \) = 1 from an external point is

2
4
6
5

If B, B¹ are the ends of minor axis, F₁ ,F₂ are foci of the ellipse \(\frac { { x }^{ 2 } }{ 8 } +\frac { { y }^{ 2 } }{ 4 } \) = 1 then area of F₁BF₂B¹ is

16
8
16\(\sqrt2\)
32\(\sqrt2\)

The tangent at any point P on the ellipse \(\frac { { x }^{ 2 } }{ 6 } +\frac { { y }^{ 2 } }{ 3 } \) = 1 whose centre C meets the major axis at T and PN is the perpendicular to the major axis; The CN CT = ______________

\(\sqrt6\)
3
\(\sqrt3\)
6

If t₁ and t₂ are the extremities of any focal chord of y² = 4ax then t₁t₂is ______________

-1
0
±1
\(\frac12\)

The point of curve y = 2x²- 6x - 4 at which the targent is parallel to x - axis is

\(\left( \frac { 5 }{ 2 } ,\frac { -7 }{ 12 } \right) \)
\(\left( \frac { -5 }{ 2 } ,\frac { -7 }{ 2 } \right) \)
\(\left( \frac { -5 }{ 2 } ,\frac { 17 }{ 12 } \right) \)
\(\left( \frac { 3 }{ 2 } ,\frac { -17 }{ 2 } \right) \)

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