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From the perfect curve of a football’s path to the elliptical orbits of planets, conic sections are all around us. These curves- circle, ellipse, parabola, and hyperbola are formed by slicing a cone at different angles. In mathematics, they will help you to model motion, design structures and even focus light and sound! In this exercise, you will apply all the key concepts of conic sections, like equations of circles, ellipses, parabolas, hyperbolas, etc. The questions are a blend of geometry and algebra that will test your overall understanding of the chapter.
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The NCERT is a key guide that offers step-by-step calculations and clear explanations to help you tackle a variety of problems. These NCERT solutions will help you improve your speed and accuracy and will prepare you for upcoming exams.
Question 1: If a parabolic reflector is 20 cm in diameter and 5 cm deep, find the focus.
Answer:
The parabolic reflector opens towards the right.
So the equation of a parabolic reflector will be,
Now, since this curve will pass through the point (5,10) if we assume origin at the optical centre,
So
Hence, the focus of the parabola is,
Alternative Method,
As we know, on any concave curve
Hence, Focus
Hence, the focus is 5 cm right to the optical centre.
Answer:
Since the Axis of the parabola is vertical, let the equation of the parabola be,
it can be seen that this curve will pass through the point (5/2, 10) if we assume the origin at the bottom end of the parabolic arch.
So,
Hence, the equation of the parabola is
Now, when y = 2 the value of x will be
Hence, the width of the arch at this height is
Answer:
Given,
The width of the parabolic cable = 100m
The length of the shorter supportive wire attached = 6m
The length of the longer supportive wire attached = 30m
Since the rope opens upwards, the equation will be of the form
Now, if we consider the origin at the centre of the rope, the equation of the curve will pass through points, (50,30-6)=(50,24)
Hence, the equation of the parabola is
Now, at a point 18 m right from the centre of the rope, the x coordinate of that point will be 18, so by the equation, the y coordinate will be
Hence, the length of the supporting wire attached to the roadway from the middle is 3.11+6=9.11m.
Answer:
The equation of the semi-ellipse will be of the form
Now, according to the question,
the length of major axis = 2a = 8
The length of the semimajor axis =2
Hence, the equation will be,
Now, at point 1.5 cm from the end, the x coordinate is 4-1.5 = 2.5
So, the height at this point is
Hence, the height of the required point is 1.56 m.
Answer:
Let
Now, at a point 3 cm from the end,
At the point touching the ground
Now, as we know the trigonometric identity,
Hence, the equation is,
Answer:
Given the parabola,
Comparing this equation with
Now, as we know, the coordinates of the ends of the latus rectum are
So, the coordinates of the latus rectum are,
Now the area of the triangle with coordinates (0,0),(6,3) and (-6,3)
Width of the triangle = 2*6=12
Height of the triangle = 3
So the area =
Hence, the required area is 18 units square.
Answer:
As we know, if a point moves in a plane in such a way that its distance from two points remains constant, then the path is an ellipse.
Now, according to the question,
The distance between the point where the sum of the distances from a point is constant = 10
Now, the distance between the foci=8
Now, as we know the relation,
Hence, the equation of the ellipse is,
Hence, the path of the man will be
Answer:
Given an equilateral triangle inscribed in a parabola with the equation.
One coordinate of the triangle is A(0,0).
Now, let the other two coordinates of the triangle be
Now, since the triangle is equilateral,
The coordinates of the points of the equilateral triangle are,
So, the side of the triangle is
Also read
1. Circle
A circle is the set of all points in a plane that are equidistant from a fixed point called the center.
where
2. Parabola
The collection of all points that are equally spaced between a given line and a fixed point (focus) is called a parabola.
Standard equation (horizontal axis) is given by
3. Ellipse
An ellipse is the set of all points in a plane such that the sum of distances from two fixed points (called foci) is constant.
4. Hyperbola
The collection of all points in a plane where the distance difference between two fixed points (foci) is constant is called a hyperbola.
5. Latus Rectum
A line segment that runs through the focus and is perpendicular to a conic's axis of symmetry is called the latus rectum. It helps in defining the width of a conic near the focus.
Also read
Students can download the NCERT solutions from the link below and can make their studies more effective.
NCERT Solutions for Class 11 Maths |
NCERT Solutions for Class 11 Physics |
NCERT Solutions for Class 11 Chemistry |
NCERT Solutions for Class 11 Biology |
Take a step ahead in your preparations by following our NCERT exemplar solutions that are designed to give you a deeper understanding of the concepts.
NCERT Exemplar Solutions for Class 11 Maths |
NCERT Exemplar Solutions for Class 11 Physics |
NCERT Exemplar Solutions for Class 11 Chemistry |
NCERT Exemplar Solutions for Class 11 Biology |
Equation of circle =>
Equation of the parabola =>
Given equation of the parabola =>
Compare with
Coordinate of focus is (3,0).
The given equation involves
Given equation of the parabola =>
Compare with
The equation of the directrix of the parabola is x = – 3
Given equation of the parabola =>
Compare with
Length of the latus rectum is 4a = 4 × 3 = 12.
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