Q.5 a) Show that the plane $latex {2x-2y+ z+12=0}&s=1$ touches the sphere $latex {x^2+y^2+z^2-2x-4y+2z-3=0}&s=1$. Also find the point of contact. Solution : The sphere has center at the point $latex {(1,2,-1)}&s=1$ and its radius is equal to $latex {\sqrt {(1)^2 + (2)^2 + (-1)^2 - (-3)} = \sqrt {1+4+1+3} = \sqrt {9} = 3 \… Continue reading M II May 2016

# Category: M II

## Curve Tracing II

In the previous section, we learnt how to trace the curves, when equations are in Cartesian form. In this section, we will learn to trace polar curves. General Observations (For Equations in Polar Coordinates) I) The equation in polar form is generally $latex r=f(\theta)$, where $latex r^2 = x^2+y^2$ and $latex tan (\theta) = \frac… Continue reading Curve Tracing II

## Planes

Definition of Plane Let there be three non-collinear points. There exist 3 distinct lines, which pass through these points taken 2 at a time. The triangle so formed is a planar surface. A line joining any 2 points on a plane always lies on the plane. Equation of Plane : Normal Form Consider a plane.… Continue reading Planes

## Differentiation Under Integral Sign and Error Function

Differentiation under Integral Sign, DUIS Introduction Not all integrals can be evaluated using analytical techniques, such as integration by substitution, by parts or by partial fractions. People come up with different ways of solving the integrals and DUIS is one of them. The class of definite integrals can be treated as integrals of function of… Continue reading Differentiation Under Integral Sign and Error Function

## Rectification

Related : Curve Tracing The process of rectification involves computation of lengths of curves. The logic behind this is based on right triangle geometry and calculus. Consider a curve in $latex XY$ plane. Let the curve be divided into infinitesimally smaller parts. Let $latex dS$ be the length of a part. As we take the… Continue reading Rectification

## Curve Tracing I

Introduction Given an equation of a curve, say $latex y=f(x)$, the standard process of plotting involves obtaining many pairs of coordinates $latex (x,y)$, which satisfy the equation of curve. Having plotted a sufficiently large number of points, one gets a good picture of the curve. In curve tracing (or curve sketching), we do not plot… Continue reading Curve Tracing I

## Differential Equations II

Prerequisite : Differential Equations I Introduction In the previous post on differential equations, we looked at various differential equations of 1st order and 1st degree; but we had no clue about their nature, behavior and the fields of their application. Since derivatives are a tool to measure the rate of change of one quantity w.r.t.… Continue reading Differential Equations II