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Two Dimensional Analytical Geometry-II 3 Mark Book Back Question Paper With Answer Key

12th Standard

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Maths

Time : 00:45:00 Hrs
Total Marks : 60

    3 Marks

    20 x 3 = 60
  1. A line 3x+4y+10 = 0 cuts a chord of length 6 units on a circle with centre of the circle (2,1). Find the equation of the circle in general form.

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

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

  4. Find the equation of the hyperbola with vertices (0, ±4) and foci(0, ±6).

  5. Find the equation of the ellipse in each of the cases given below:
    length of latus rectum 4, distance between foci 4 \( \sqrt{ 2}\) , centre (0, 0) and major axis as y - axis.

  6. Find the equations of the tangent and normal to hyperbola 12x2−9y= 108 at \(\theta =\frac { \pi }{ 3 } \) (Hint: use parametric form)

  7. A concrete bridge is designed as a parabolic arch. The road over bridge is 40 m long and the maximum height of the arch is 15 m. Write the equation of the parabolic arch.

  8. The parabolic communication antenna has a focus at 2m distance from the vertex of the antenna. Find the width of the antenna 3m from the vertex.

  9. The equation y = \(\frac { 1 }{ 32 } \)x2 models cross sections of parabolic mirrors that are used for solar energy. There is a heating tube located at the focus of each parabola; how high is this tube located above the vertex of the parabola?

  10. An equation of the elliptical part of an optical lens system is \(\frac { { x }^{ 2 } }{ 16 } +\frac { { y }^{ 2 } }{ 9 } \) = 1. The parabolic part of the system has a focus in common with the right focus of the ellipse. The vertex of the parabola is at the origin and the parabola opens to the right. Determine the equation of the parabola.

  11. The equation of the ellipse is \(\frac { { \left( x-11 \right) }^{ 2 } }{ 484 } +\frac { { y }^{ 2 } }{ 64 } =1\). ( x and y are measured in centimeters) where to the nearest centimeter, should the patient’s kidney stone be placed so that the reflected sound hits the kidney stone?

  12. The circle passing through the points of intersection (real or imaginary) of the line lx+my+n = 0 and the circle x2 + y2 +2gx+2 fy+c =0 is the circle of the form
    x2 + y2 + 2gx + 2 fy + c +\(\lambda\) (lx + my + n) = 0 \(\lambda \in \mathbb{R}^{1}\)

  13. The equation of a circle with (x1, y1 ) and (x2, y2 ) as extremities of one of the diameters of the circle is (x − x1)(x − x2 ) + ( y − y1 )( y − y2) = 0

  14. The position of a point P(x1 , y 1)  with respect to a given circle x2 + y2 + 2gx + 2 fy + c = 0 in the plane containing the circle is outside or on or inside the circle according as
    \(x_{1}^{2}+y_{1}^{2}+2 g x_{1}+2 f y_{1}+c \text { is } \begin{cases}>0 & \text { or } \\ =0 & \text { or } \\ <0 & \end{cases}\)

  15. From any point outside the circle x2 + y2 = a2 two tangents can be drawn

  16. The sum of the focal distances of any point on the ellipse is equal to length of the major axis

  17. Three normals can be drawn to a parabola y2 = 4ax from a given point, one of which is always real.

  18. Find the equation of the ellipse in each of the cases given below:
    foci (±3, 0), e = \(\frac { 1 }{ 2 } \)

  19. Find the equation of the ellipse in each of the cases given below:
    foci (0, ±4) and end points of major axis are (0, ±5).

  20. Find the equation of the ellipse in each of the cases given below:
    length of latus rectum 8, eccentricity = \(\frac { 3 }{ 5 } \), centre (0, 0) and major axis on x -axis.

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