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CIRCLE

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 Circle A circle is a locus of a point which moves along a path such that the distance from a fixed point is always a constant. The fixed point is known as center and the fixed distance or constant is known as radius of the circle. Equation of circle with center (h, k) and radius r The equation of circle with center (h, k) and radius r is given by If the center is origin, i.e, (0, 0), then, equation of circle changes to General equation of a circle General equation of a circle is given by Radius of the circle Radius of a circle having equation    is given by Note: If    , then the circle is real. If    , then the circle is a point circle.  If    , then the circle is an imaginary circle. Some important points: For a circle Coefficient of   = Coefficient of   Coefficient of xy term in the equation of the circle is equal to zero. If two circles have same center, then they are concentric. Concyclic qua...
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TRIGONOMETRY INTRODUCTION

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 In this post, we'll learn about Trigonometric ratios of acute angles. An angle is said to be acute if it is greater than 0 ° but less than 90 °. Consider a right-angled triangle ABC with right angle at B. We know that, The side opposite to the right angle is the hypotenuse. The side opposite to angle A is called as opposite. The side adjacent to angle A is called as adjacent. Consider the triangle ABC as shown in the figure, The ratio of lengths of any two sides of the triangle can be determined by the angle A and are independent of size of the triangle. The number of such possible ratios are 6. Each of them has a name as follows, The ratios sin, cos, tan, cot, sec and cosec stand for sine, cosine, tangent, cotangent, secant, cosecant of angle A respectively. These functions of angle A are called trigonometric ratios. The trigonometric ratios of some standard angles are given below. We know that, an identity is an equation which holds true for every value of occurring variables. T...
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Heron's Formula

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  A triangle is ABC is given with the side lengths a, b, c as shown in the figure. The area of the above triangle is given by Heron's Formula which is where s is semi-perimeter of the triangle. Example: A scalene triangle has sides of lengths 7 cm, 8 cm, 9 cm. Find area of the triangle. Solution: Consider, a=7, b=8, c=9 Now, Substituting all the values in the formula, we get,
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Distance between two points

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  Distance between two points Consider two A(x1,y1), B(x2,y2) points in the coordinate system. Distance between these two points is given by the formula, We can prove the formula by using Pythagoras Theorem From Pythagoras Theorem, Square of hypotenuse is equal to sum of squares of sides. Therefore, we get, Example: Find the distance between the points (1,2) and (-4,-3) Solution: Distance is given by
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Straight Line

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  Straight Line A Straight line is a line joining two points. Suppose there are two points A(x1,y1) and B(x2,y2) in the coordinate system. Then, the Equation of the line joining A and B is given by y-y1=m(x-x1) where m is the slope of the line AB. m is given by m=(y2-y1)/(x2-x1). An example of straight line is given below. Example: There are two points namely P(1,2) and Q(3,-8). Give the equation of the line joining the two points. Solution: As discussed above,  The line joining two points is given by y-y1=m(x-x1) First, we need to calculate slope of the line m=(-8-2)/(3-1)=-10/2=-5 => y-2=-5(x-1) => y-2=-5x+5 => 5x+y-7=0 Therefore, Equation of the line joining P and Q is given by  5x+y-7=0 General equation of a straight line is given by y=ax+b Condition for the lines to be parallel If the two lines    and    are parallel, then Condition for the lines to be perpendicular If the two lines    and    are p...
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