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12th Maths Weekly Test -3 (Two Dimensional Analytical Geometry-II)

St. Britto Hr. Sec. School - Madurai

12th Maths Weekly Test -3 (Two Dimensional Analytical Geometry-II)-Aug 2020

Time : 1.30 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

Part - B

Generic 2 - Answer All Question

Find the equation of the circle with centre (2,-1) and passing through the point (3,6) in
standard form.

Find the equation of the circle with centre (2,3) and passing through the intersection of the
lines 3x−2y−1=0and 4x+y−27=0.

Find the equation of the circle through the points (1,0),(-1,0) , and (0,1) .

Ify=\( 2\sqrt { 2 } \)x +c is a tangent to the circlex²+y²=16 , find the value of c .

Determine whether the points(-2,1),(0,0) and (-4,-3) lie outside, on or inside the circle
x²+y²−5x+2y−5=0 .

If the equation3x²+(3−p)xy+qy²−2px =8pq represents a circle, find p and q . Also determine the centre and radius of the circle

Find the equation of the ellipse in each of the cases given below:
(i) foci(±3 0),e =\(\frac { 1 }{ 2 } \)
(ii) foci (0,±4)and end points of major axis are(0,±5).
(iii) length of latus rectum 8, eccentricity =\(\frac { 3 }{ 5 } \) and major axis on x -axis.
(iv) length of latus rectum 4 , distance between foci4 \( \sqrt{ 2}\) and major axis as y - axis.

Find the vertex, focus, equation of directrix and length of the latus rectum of the following:
y² =16x

Prove that the length of the latus rectum of the hyperbola \(\frac { { x }^{ 2 } }{ { a }^{ 2 } } -\frac { { y }^{ 2 } }{ { b }^{ 2 } } \) = 1 is \(\frac { { 2b }^{ 2 } }{ a } \).

Identify the type of conic and find centre, foci, vertices, and directrices of each of the following :
(i)\(\frac { { \left( x-3 \right) }^{ 2 } }{ 225 } +\frac { { \left( y-4 \right) }^{ 2 } }{ 289 } =1\)
(ii)\(\frac { { \left( x+1 \right) }^{ 2 } }{ 100 } +\frac { { \left( y-2 \right) }^{ 2 } }{ 64 } =1\)
(iii)\(\frac { { \left( x+3 \right) }^{ 2 } }{ 225 } -\frac { { \left( y-4 \right) }^{ 2 } }{ 64 } =1\)
(iv)\(\frac { { \left( y-2 \right) }^{ 2 } }{ 25 } \frac { { \left( x+1 \right) }^{ 2 } }{ 16 } =1\)
(v)18x²+12y²−144x+48y+120 = 0
(vi) 9x²−y²−36x−6y+18 = 0

Part - C

Generic 3 - Answer All Question

Find the circumference and area of the circle x² +y² - 2x + 5y + 7 = 0

Find the condition for the line lx + my + n = 0 is Rtangent to the circle x² + y² = a²

Find the equatlon of the ellipse whose e = \(\frac34\), foci ony-axl ,centre at origin and passing through (6,4).

Find the equation of the hyperbola whose conjugate axis is 5 and the distance between the foci is 13.

Find the value of c if y = x + c is a tangent to the hyperbola 9x² - 16y² = 144.

Find the equation of the circle described on the chord 3x+y+5= 0 of the circle x²+y²=16 as diameter.

A circle of radius 3 units touches both the axes. Find the equations of all possible circles formed in the general form.

Find the equations of the tangent and normal to the circle x²+y²=25 at P(-3,4).

Find the length of Latus rectum of the ellipse\(\frac { { x }^{ 2 } }{ { a }^{ 2 } } +\frac { { y }^{ 2 } }{ { b }^{ 2 } } =1\)

Find the equation of the parabola whose vertex is (5,-2) and focus(2,-2) .

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