**The Standard Form of Circle**

- (x – h)
^{2}+ (y – k)^{2}= r^{2}

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:
**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**

Correct Answer is Option (c)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^{}^{}

**a.**Centre (4, 3) & Radius = 5 units**b.**Centre (4, 3) & Radius = 25 units**c.**Centre (-4, -3) & Radius = 25 units**d.**Centre (-4, -3) & Radius = 5 units

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**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.

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

**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.

**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.**

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:**

As (t, 1) lies inside the circle, so its distance from centre i.e. origin should be less than radius i.e.

**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 asx = 3&y = – 3.

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

⇒ Slope of line m = =– 7So, the equation of line is:

⇒ y + 3 = – 7 (x – 3)

⇒ 7x + y – 18 = 0Alternate 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 =So, the equation of line is: