## Lecture 7 Miscellaneous Exercise Chapter 11

 MISCELLANEOUS EXERCISE

Question 4. An arch is in the form of a semi-ellipse. It is 8 m wide and 2 m high at the center. Find the height of the arch at a point 1.5 m from one end.

Question 5. A rod of length 12 cm moves with its ends always touching the coordinate axes. Determine the equation of the locus of a point P on the rod, which is 3 cm from the end in contact with the x-axis.

Question 6. Find the area of the triangle formed by the lines joining the vertex of the parabola $x^2=12y$ to the ends of its latus rectum.

Question 7. A man running a racecourse notes that the sum of the distances from the two flag posts from him is always 10 m and the distance between the flag posts is 8 m. Find the equation of the posts traced by the man.

Question 8. An equilateral triangle is inscribed in the parabola $y^2=4ax$, where one vertex is at the vertex of the parabola. Find the length of the side of the triangle.

## Lecture 6 Hyperbolas Conic Sections Class 11 Maths

Explanation and method to remember all rules of standard hyperbolas

 NCERT EXERCISE 11.4

In each of the Exercises 1 to 6, find the coordinates of the foci and the vertices, the eccentricity and the length of the latus rectum of the hyperbolas.

Question 1. $\frac{x^2}{16}-\frac{y^2}{9}=1$

Question 2. $\frac{y^2}{9}-\frac{x^2}{27}=1$

Question 3. $9y^2-4x^2=36$

Question 4. $16x^2-9y^2=576$

Question 5. $5y^2-9x^2=36$

Question 6. $49y^2-16x^2=784$

In each of the Exercises 7 to 15, find the equations of the hyperbola satisfying the given conditions.

Question 7. Vertices (± 2, 0), foci (± 3, 0)

Question 8. Vertices (0, ± 5), foci (0, ± 8)

Question 9. Vertices (0, ± 3), foci (0, ± 5)

Question 10. Foci (± 5, 0), the transverse axis is of length 8.

Question 11. Foci (0, ±13), the conjugate axis is of length 24.

Question 12. Foci $(\pm 3 \sqrt{5}, 0)$, the latus rectum is of length 8.

Question 13. Foci (± 4, 0), the latus rectum is of length 12

Question 14. vertices (± 7,0), $e = \frac{4}{3}$

Question 15. Foci $(0, \pm \sqrt{10})$, passing through (2, 3)

## Class 11 Maths Conic Sections Chapter 11 Lecture 5

Topics Discussed:

Meaning and definition of Hyperbola
Difference between circle, parabola, ellipse and hyperbola
Geometrical Verification of definition of hyperbola
Real life examples of Hyperbolas
Transverse axis and conjugate axis of Hyperbola
Introduction to a, b and c of hyperbolas
Derivation for equation of standard hyperbola

## Lecture 4 Ellipse Conic Sections Class 11 Maths

Explanation of all basic details of standard ellipse, method to remember their rules

 NCERT Exercise 11.3

In each of the Exercises 1 to 9, find the coordinates of the foci, the vertices, the length of major axis, the minor axis, the eccentricity and the length of the latus rectum of the ellipse.

Question 1.  $\frac{x^2}{36}+\frac{y^2}{16}=1$

Question 2. $\frac{x^2}{4}+\frac{y^2}{25}=1$

Question 3. $\frac{x^2}{16}+\frac{y^2}{9}=1$

Question 4. $\frac{x^2}{25}+\frac{y^2}{100}=1$

Question 5. $\frac{x^2}{49}+\frac{y^2}{36}=1$

Question 6. $\frac{x^2}{100}+\frac{y^2}{400}=1$

Question 7. $36x^2+4y^2=144$

Question 8. $16x^2+y^2=16$

Question 9. $4x^2+9y^2=36$

In each of the following Exercises 10 to 20, find the equation for the ellipse that satisfies the given conditions:

Question 10. Vertices (± 5, 0), foci (± 4, 0)

Question 11. Vertices (0, ± 13), foci (0, ± 5)

Question 12. Vertices (± 6, 0), foci (± 4, 0)

Question 13. Ends of major axis (± 3, 0), ends of minor axis (0, ± 2)

Question 14. Ends of major axis $(0, \pm \sqrt{5})$, ends of minor axis (± 1, 0)

Question 15. Length of major axis 26, foci (± 5, 0)

Question 16. Length of minor axis 16, foci (0, ± 6).

Question 17. Foci (± 3, 0), a = 4

Question 18. b = 3, c = 4, center at the origin; foci on the x axis.

Question 19. Center at (0,0), major axis on the y-axis and passes through the points (3, 2) and (1, 6).

Question 20. Major axis on the x-axis and passes through the points (4, 3) and (6, 2).

## Ellipse Chapter 11 Conic Sections Lecture 3

Topics discussed:

What is an ellipse with animation, Real life examples of ellipse, Major axis and minor axis, Derivations for relationship between a, b and c of ellipse, Derivation for equation of ellipse, Derivation for latus rectum of ellipse, Eccentricity of ellipse

## Parabola Chapter 11 Conic Sections Lecture 2

Real life examples of parabola, Standard forms of Parabola, Definition and geometric proof for parabola, Derivation for equation of parabola, Way to remember all rules of parabolas, Derivation for length of latus rectum of parabola

 NCERT EXERCISE 11.2

In each of the following Exercises 1 to 6, find the coordinates of the focus, axis of the parabola, the equation of the directrix and the length of the latus rectum.

Question 1. $y^2 = 12x$

Question 2. $x^2 = 6y$

Question 3. $y^2 = -8x$

Question 4. $x^2=-16y$

Question 5. $y^2 = 10x$

Question 6. $x^2=-9y$

In each of the Exercises 7 to 12, find the equation of the parabola that satisfies the given conditions:

Question 7. Focus (6,0); directrix x = – 6

Question 8. Focus (0,–3); directrix y = 3

Question 9. Vertex (0, 0); focus (3, 0)

Question 10. Vertex (0, 0); focus (–2, 0)

Question 11. Vertex (0, 0) passing through (2, 3) and axis is along x-axis.

Question 12. Vertex (0, 0), passing through (5, 2) and symmetric with respect to y-axis.

 MISCELLANEOUS EXERCISE

Question 1.If a parabolic reflector is 20 cm in diameter and 5 cm deep, find the focus.

Question 2. An arch is in the form of a parabola with its axis vertical. The arch is 10 m high and 5 m wide at the base. How wide is it 2 m from the vertex of the parabola?

Question 3.The cable of a uniformly loaded suspension bridge hangs in the form of a parabola. The roadway which is horizontal and 100 m long is supported by vertical wires attached to the cable, the longest wire being 30 m and the shortest being 6 m. Find the length of a supporting wire attached to the roadway 18 m from the middle.

## Circles Chapter 11 Conic Sections Class 11 Maths

How to study Conic Sections in Class 11, Introduction to Circle, Parabola, Ellipse and Hyperbola, Animation to understand why they are called conic sections, How to derive equation of circle

 NCERT EXERCISE 11.1

In each of the following Exercises 1 to 5, find the equation of the circle with

Question 1. center (0, 2) and radius 2

Question 2. center (–2, 3) and radius 4

Question 3. center $\left ( \frac{1}{2}, \frac{1}{4} \right )$ and radius $\frac{1}{12}$

Question 4. center (1, 1) and radius $\sqrt{2}$

Question 5. center $(-a, -b)$ and radius $\sqrt{a^2-b^2}$

In each of the following Exercises 6 to 9, find the center and radius of the circles.

Question 6. $(x+5)^2+(y-3)^2=36$

Question 7. $x^2+y^2-4x-8y-45=0$

Question 8. $x^2+y^2-8x+10y-12=0$

Question 9. $2x^2+2y^2-x=0$

Question 10. Find the equation of the circle passing through the points (4, 1) and (6, 5) and whose center is on the line $4x + y = 16$.

Question 11. Find the equation of the circle passing through the points (2, 3) and (–1, 1) and whose center is on the line $x – 3y – 11 = 0$.

Question 12. Find the equation of the circle with radius 5 whose centre lies on x-axis and passes through the point (2, 3).

Question 13. Find the equation of the circle passing through (0, 0) and making intercepts a and b on the coordinate axes.

Question 14. Find the equation of a circle with center (2, 2) and passes through the point (4, 5).

Question 15. Does the point (–2.5, 3.5) lie inside, outside or on the circle $x^2 + y^2 = 25$?