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 \(ax + b{y^2} + c = 0\)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.