## 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 $$ax + b{y^2} + c = 0$$i.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.