Q83.The length of the latus rectum and directrices of a hyperbola with eccentricity e are 9 and x = ± 4 , √13 respectively. Let the line y −√3x + √3 = 0 touch this hyperbola at (x0, y0). If m is the product of the focal distances of the point (x0, y0), then 4e2 + m is equal to ________
What This Question Tests
This multi-concept question integrates several hyperbola properties including latus rectum, directrices, eccentricity, and tangent conditions. It requires careful algebraic manipulation to combine these concepts.
Concepts Tested
Formulas Used
Latus Rectum = 2b²/a
Directrices x = ±a/e
b² = a²(e²-1)
Tangent y = mx + c to x²/a² - y²/b² = 1 implies c² = a²m² - b²
Product of focal distances = |e²x₀² - a²|
📚 NCERT Sections This Tests
9.23 — (A) At What Distance Should The Lens Be Held From The Card Sheet In
Physics Class 12 · Chapter 9
9.23 (a) At what distance should the lens be held from the card sheet in Exercise 9.22 in order to view the squares distinctly with the maximum possible magnifying power? (b) What is the magnification in this case? (c) Is the magnification equal to the magnifying power in this case? Explain.
9.8 — A Beam Of Light Converges At A Point P. Now A Lens Is Placed In The
Physics Class 12 · Chapter 9
9.8 A beam of light converges at a point P. Now a lens is placed in the path of the convergent beam 12cm from P. At what point does the beam converge if the lens is (a) a convex lens of focal length 20cm, and (b) a concave lens of focal length 16cm?
9.9 — An Object Of Size 3.0Cm Is Placed 14Cm In Front Of A Concave Lens Of
Physics Class 12 · Chapter 9
9.9 An object of size 3.0cm is placed 14cm in front of a concave lens of focal length 21cm. Describe the image produced by the lens. What happens if the object is moved further away from the lens? 249 Reprint 2025-26 Physics 9.10 What is the focal length of a convex lens of focal length 30cm in contact with a concave lens of focal length 20cm? Is the system a converging or a diverging lens? Ignore thickness of the lenses. 9.11 A compound microscope consists of an objective lens of focal length 2.0 cm and an eyepiece of focal length 6.25 cm separated by a distance of 15cm. How far from the objective should an object be placed in order to obtain the final image at (a) the least distance of distinct vision (25cm), and (b) at infinity? What is the magnifying power of the microscope in each case? 9.12 A person with a normal near point (25 cm) using a compound microscope with objective of focal length 8.0 mm and an eyepiece of focal length 2.5cm can bring an object placed at 9.0mm from the objective in sharp focus. What is the separation between the two lenses? Calculate the magnifying power of the microscope, 9.13 A small telescope has an objective lens of focal length 144cm and an eyepiece of focal length 6.0cm. What is the magnifying power of the telescope? What is the separation between the objective and the eyepiece? 9.14 (a) A giant refracting telescope at an observatory has an objective lens of focal length 15m. If an eyepiece of focal length 1.0cm is used, what is the angular magnification of the telescope? (b) If this telescope is used to view the moon, what is the diameter of the image of the moon formed by the objective lens? The diameter of the moon is 3.48 × 106m, and the radius of lunar orbit is 3.8 × 108m. 9.15 Use the mirror equation to deduce that: (a) an object placed between f and 2f of a concave mirror produces a real image beyond 2f. (b) a convex mirror always produces a virtual image independent of the location of the object. (c) the virtual image produced by a convex mirror is always diminished in size and is located between the focus and the pole. (d) an object placed between the pole and focus of a concave mirror produces a virtual and enlarged image. [Note: This exercise helps you deduce algebraically properties of images that one obtains from explicit ray diagrams.] 9.16 A small pin fixed on a table top is viewed from above from a distance of 50cm. By what distance would the pin appear to be raised if it is viewed from the same point through a 15cm thick glass slab held parallel to the table? Refractive index of glass = 1.5. Does the answer depend on the location of the slab? 9.17 (a) Figure 9.28 shows a cross-section of a ‘light pipe’ made of a glass fibre of refractive index 1.68. The outer covering of the pipe is made of a material of refractive index 1.44. What is the range of the angles of the incident rays with the axis of the pipe for which total reflections inside the pipe take place, as shown in the figure. FIGURE 9.28 Reprint 2025-26 Ray Optics and Optical Instruments (b) What is the answer if there is no outer covering of the pipe?
📋 Question Details
- Chapter
- Hyperbola
- Topic
- Properties of Hyperbola and Tangents
- Year
- 2024
- Shift
- 06 Apr Shift 2
- Q Number
- Q83
- Type
- Multi concept
- NCERT Ref
- Class 11 Mathematics Ch 11: Conic Sections
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