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**1. Standard form of Circle**

- (x - h)
^{2}+ (y - k)^{2}= r^{2}

Where the centre is (h, k) and radius is ‘r’. - If centre of the circle is at the origin and radius is 'r', then the equation of the circle is:
**x**^{2}+ y^{2}= r^{2} - 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**

**Answer. (c) 32****Solution. **Since the line segment joining (4, 4) and (0, 0) is a radius of the circle:

r^{2} = 4^{2} + 4^{2} = 32

Therefore, area = πr^{2} = 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^{}^{}

(x - 4)

View Solution

- 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.

Ellipse - The equation of an ellipse centered at the origin and with an axial intersection at
**(±, a, 0)**and**(± b, 0)**is:

**3. 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.

Parabola - You should know, however, that the graph of the general quadratic equation:

y = ax^{2}+ 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 = ax**is a parabola.^{2}+ bx + c**Examples:**y = x^{2}- 2x + 1 and y = - x^{2}- 4 are examples of some parabolic equations.

**4. The Hyperbola**

- If a right circular cone is intersected by a plane parallel to its axis, part of a hyperbola is formed.

Hyperbola - The equation of a hyperbola at the origin and with foci on the x-axis is:

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

The area of the square is = **32 sq. units**.

**Example.3 If point (t, 1) lies inside circle x ^{2} + y^{2 }= 10, then t must lie between:**

4x - 3y - 21 = 0 …..(1)

3x - y - 12 = 0 ….(2)

Solving (1) and (2), we get point of intersection as

Now we have two points (3, -3) & (2, 4)

⇒ Slope of line m = =

⇒ y + 3 = - 7 (x - 3)

⇒ 7x + y - 18 = 0

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 =

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