## 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.

• 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.