What to find : Locus of point from which tangents drawn to a circle have a particular angle (Angle not equal to zero) between them. Hello, Everyone. Before going to find our required solution, let us try to observe a phenomenon in the life. Figure : Illustration of variation of the angle with perpendicular distance from the wall Assumptions : Let us assume that a person X is standing in front of a wall shown in above figure. Assume height of the wall is more than the person. Let A, B and C be the different positions of eye of the person as the person moves on the horizontal line segment shown in the above figure. Let "O" be the point on the wall lying on the horizontal line segment drawn from the eye to the wall such that it is perpendicular to the wall. Let "E" be the one end of the wall as shown in the above figure. In the above figure, variation of the angle means variation of the a ngle between the horizontal line segment and line segment joining the eye ...
Hello Everyone. Before going to the proof, let us go into the basics of Rational and Irrational numbers. Rational Number : A number that can be expressed as the fraction p/q of two integers p and q, where q not equal to zero. Irrational Number : A number that cannot be expressed as the fraction p/q of two integers p and q, where q not equal to zero. What is required to prove : Sum of a rational number and an irrational number will be an irrational number. Proof : Let "m" be an irrational number. --------- (1) Let "n" be a rational number. ------------(2) Let us think that we don't know whether m + n is rational or irrational number. So, let us assume m + n as a rational number initially. Then, m + n = p/q where p and q are some integers and q is not equal to zero. (According to the definition) So, m + n = p/q Subtract "n" on both sides. Then, m = (p/q) - n = difference of two rational numbers Difference of two rational numbers will be a...
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