How to Find the Equation of a Perpendicular Line: Step-by-Step Guide

1. Determine the Slope of the Given Line: To begin, identify the slope \((m)\) of the given line. 

2. Calculate the Negative Reciprocal of the Slope: Find the negative reciprocal of the slope, denoted as \(\frac{-1}{m}\). This value represents the slope of the perpendicular line. 

3. Identify a Point on the Perpendicular Line: Select a point that lies on the perpendicular line. 

You can either use a provided point or choose a convenient one. 

4. Use the Point-Slope Form to Write the Equation: Utilize the point-slope form of a line, which is y - y₁ = m(x - x₁). Substitute the negative reciprocal of the slope \((\frac{-1}{m})\) for m and the coordinates of the point (x₁, y₁) into the equation. 

How to Find the Equation of an Ellipse with Ease

Unveiling the Secrets 

Introduction: Understanding and working with ellipses is an essential skill for mathematicians, physicists, and engineers alike. Whether you're analyzing planetary orbits or designing architectural structures, knowing how to find the equation of an ellipse is a valuable tool. In this article, we will delve into the step-by-step process of determining the equation of an ellipse, unraveling its mathematical intricacies along the way. 

Permutation Without Repetition

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1. In how many ways can the letters of the word "APPLE" be arranged?

2. How many different ways can the numbers 1, 2, 3, 4, 5 be arranged?

3. In how many ways can a committee of 3 members be formed from a group of 6 people?

4. A box contains 6 red balls and 4 blue balls. How many different ways can you arrange the balls?

5. In how many ways can the letters of the word "BANANA" be arranged?

Metric Units Conversion Chart

Length:

1 millimeter (mm) = 0.1 centimeters (cm)

1 centimeter (cm) = 10 millimeters (mm)

1 meter (m) = 100 centimeters (cm) = 1,000 millimeters (mm)

1 kilometer (km) = 1,000 meters (m)

Imperial Units Conversion Chart

Length:

1 inch = 2.54 centimeters

1 foot = 12 inches

1 yard = 3 feet

1 mile = 1,760 yards

Combinations

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1. In a game, there are 10 players, and the coach needs to choose a captain and a vice-captain. How many different combinations of captain and vice-captain are possible?
   
   
2. A school has 5 different clubs, and each student can join only one club. If there are 25 students in the school, how many different ways are there for the students to join the clubs?

Equation of parabola from graph

Transformed equation of parabola is: \(y=af(x-h)^2+k\) \(……(1)\)

Where \(a=\)Vertical stretch or compression, \(h=\)horizontal shift (left or right) \(k=\)Vertical shift (up or down) From graph, \(h=3\) and \(k=-2\); plug \(h\) and \(k\) values in eqn \((1)\) \(y=a(x-3)^2-2\) /(……(2)\) To find \(‘a’\), use the given point \((1,0)\) as: \(0=a(5-3)^2-2\) \(0=4a-2\) \(a=1/2\);plug in eqn.(2) \(y=1/2 (x-3)^2-2\);which is the reqd.eqn.