Circle, Ellipse, Parabola & Hyperbola: Examples (with Solutions) Notes | Study UPSC CSAT Preparation - UPSC

UPSC: Circle, Ellipse, Parabola & Hyperbola: Examples (with Solutions) Notes | Study UPSC CSAT Preparation - UPSC

The document Circle, Ellipse, Parabola & Hyperbola: Examples (with Solutions) Notes | Study UPSC CSAT Preparation - UPSC is a part of the UPSC Course UPSC CSAT Preparation.
All you need of UPSC at this link: UPSC

The standard form of Circle
Circle, Ellipse, Parabola & Hyperbola: Examples (with Solutions) Notes | Study UPSC CSAT Preparation - UPSC

  • (x - h)2 + (y - k)2 = r2
    Where the center is (h, k) and radius is ‘r’.
  • If center of the circle is at the origin and the radius is 'r', then the equation of the circle is:
    x2 + y2 = r2
  • This is also known as the simplest form.

Example 1: If the area of the circle shown below is kπ, what is the value of k?
(a) 4 
(b) 16
(c) 32 
(d) 20

Circle, Ellipse, Parabola & Hyperbola: Examples (with Solutions) Notes | Study UPSC CSAT Preparation - UPSC

Correct Answer is Option (c)

Since the line segment joining (4, 4) and (0, 0) is a radius of the circle:
r2 = 42 + 42 = 32
Therefore, area = πr2 = 32π ⇒ k = 32 


Try yourself:What is the coordinates of the centre and the radius of the circle with equation: 
(x - 4)2 + (y - 3)2 = 25 
View Solution


The Ellipse

  • Though not so simple as the circle, the ellipse is nevertheless the curve most often "seen" in everyday life. The reason is that every circle, viewed obliquely, appears elliptical.
    EllipseEllipse
  • The equation of an ellipse centered at the origin and with an axial intersection at (±, a, 0) and (± b, 0) is:
    Circle, Ellipse, Parabola & Hyperbola: Examples (with Solutions) Notes | Study UPSC CSAT Preparation - UPSC


The Parabola

  • When a baseball is hit into the air, it follows a parabolic path. There are all kinds of parabolas, and there’s no simple, general parabola formula for you to memorize.
    ParabolaParabola
  • You should know, however, that the graph of the general quadratic equation:
    y = ax2 + bx + c is a parabola. 
  • It’s one that opens up either on top or on the bottom, with an axis of symmetry parallel to the y-axis. The graph of the general quadratic equation y = ax2 + bx + c is a parabola.
    Examples: y = x2 - 2x + 1 and y = - x2 - 4 are examples of some parabolic equations.


The Hyperbola

  • If a right circular cone is intersected by a plane parallel to its axis, part of a hyperbola is formed.
    HyperbolaHyperbola
  • The equation of a hyperbola at the origin and with foci on the x-axis is:
    Circle, Ellipse, Parabola & Hyperbola: Examples (with Solutions) Notes | Study UPSC CSAT Preparation - UPSC


Example 2: Find the area enclosed by the figure | x | + | y | = 4.

The four possible lines are:
x + y = 4; x - y = 4; - x - y = 4 and -x + y = 4
The four lines can be represented on the coordinates axes as shown in the figure. Thus a square is formed with the vertices as shown. The side of the square is: Circle, Ellipse, Parabola & Hyperbola: Examples (with Solutions) Notes | Study UPSC CSAT Preparation - UPSC
The area of the square is Circle, Ellipse, Parabola & Hyperbola: Examples (with Solutions) Notes | Study UPSC CSAT Preparation - UPSC= 32 sq. units.

Circle, Ellipse, Parabola & Hyperbola: Examples (with Solutions) Notes | Study UPSC CSAT Preparation - UPSC


Example 3: If point (t, 1) lies inside circle x2 + y2  = 10, then t must lie between:

As (t, 1) lies inside the circle, so its distance from centre i.e. origin should be less than radius i.e.Circle, Ellipse, Parabola & Hyperbola: Examples (with Solutions) Notes | Study UPSC CSAT Preparation - UPSC
Circle, Ellipse, Parabola & Hyperbola: Examples (with Solutions) Notes | Study UPSC CSAT Preparation - UPSC


Example.4 Find the equation of line passing through (2, 4) and through the intersection of line 4x - 3y - 21 = 0 and 3x - y - 12 = 0?

4x - 3y - 21 = 0 …..(1)
3x - y - 12 = 0 ….(2)
Solving (1) and (2), we get point of intersection as x = 3 & y = - 3.
Now we have two points (3, -3) & (2, 4)
⇒ Slope of line m = Circle, Ellipse, Parabola & Hyperbola: Examples (with Solutions) Notes | Study UPSC CSAT Preparation - UPSC = - 7
So, the equation of line is:
⇒ y + 3 = - 7 (x - 3)        
⇒ 7x + y - 18 = 0
Alternate Method:
Equation of line through intersection of 4x - 3y - 21 = 0 and 3x - y - 12 = 0 is:
(4x - 3y - 21) + k(3x - y - 12) = 0.
As this line passes through (2, 4):
⇒ (4 × 2 - 3 × 4 - 21) + k(3 × 2 - 4 - 12) = 0  
⇒ k = Circle, Ellipse, Parabola & Hyperbola: Examples (with Solutions) Notes | Study UPSC CSAT Preparation - UPSC
So, the equation of line is:

Circle, Ellipse, Parabola & Hyperbola: Examples (with Solutions) Notes | Study UPSC CSAT Preparation - UPSC

The document Circle, Ellipse, Parabola & Hyperbola: Examples (with Solutions) Notes | Study UPSC CSAT Preparation - UPSC is a part of the UPSC Course UPSC CSAT Preparation.
All you need of UPSC at this link: UPSC
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