## Lecture - 1 Application Of Integrals Class 12 Maths

**LINEAR EQUATIONS**

Standard Form: Ax + By + C = 0

• It always forms a straight line.

• Find slope of line using y = mx + c form to know whether the inclination of line is towards left, right, vertical or horizontal.

o If slope of line is zero, then line is parallel to x-axis.

o If slope of line is \infty , then line is parallel to y-axis.

o If slope of line is positive real value, then line inclined from right.

o If slope of line is negative real value, then line inclined from left.

• A line in y = mx form always passes through origin.

**QUADRATIC EQUATIONS**

• It always forms a curve.

• In case of parabolas, standard form is either a{x^2} + by + c = 0 or ax + b{y^2} + c = 0i.e., only one variable with power 2.

o {y^2} = 4ax \to Parabola along positive x-axis

o {y^2} = – 4ax \to Parabola along negative x-axis

o {x^2} = 4ay \to Parabola along positive y-axis

o {x^2} = – 4ay \to Parabola along negative y-axis

• In case of circle, standard form is either {(x – a)^2} + {(y – b)^2} = {r^2} or k{x^2} + k{y^2} = c i.e., coefficient of {x^2}and {y^2}are always same.

• In case of ellipse, there are two cases:

o \frac{{{x^2}}}{{{a^2}}} + \frac{{{y^2}}}{{{b^2}}} = 1 \to Horizontal ellipse

o \frac{{{x^2}}}{{{b^2}}} + \frac{{{y^2}}}{{{a^2}}} = 1 \to Vertical ellipse

o In case it is in m{x^2} + n{y^2} = c form (i.e., coefficient of {x^2}and {y^2}are always different), first convert it into above two standard forms.