# equation of locus

According to the condition, PA = PB. \end{aligned}y20y​=x2+(y−2a)2=x2−4ay+4a2=4ax2​+a,​, Note that if the point did lie on the line, e.g. Find the equation of the locus of point P, which is equidistant from A and B. I have tried and tried to answer but it seems that I didn't get the answer. Now, the distance of a point from the X axis is its y-coordinate. That’s it for this part. Find an equation for the set of all points (x,y) satisfying the given condition: The product of its distances from the coordinate axes is 4. answer: xy= plus or minus 4 Please show how you have come up with your answer. A locus is a set of all the points whose position is defined by certain conditions. Firstly, writing the characteristic equation of the above system, So, from the above equation, we get, s = 0, -5 and -10. I need your help. AAA and BBB are two points in R2\mathbb{R}^2R2. Example – 37: Find the equation of locus of a point such that the sum of its distances from co-ordinate axes is thrice its distance from the origin. $$\sqrt{(x-1)^2+(y-1)^2}=\sqrt{(x-2)^2+(y-4)^2}$$. I’ll again split it into two parts due to its length. \end{aligned}PA2+PB2(x+a)2+y2+(x−a)2+y22x2+2y2+2a2x2+y2​=c2=c2=c2=2c2​−a2.​. Going in the reverse order, the equation y = 5 is the equation of the locus / curve, every point on which has the y -coordinate as 5 , or every point being at a distance of 5 units from the X -axis (the condition which was initially given). 4d_1^2d_2^2 &= c^4 - 2c^2\big(d_1^2+d_2^2\big) + \big(d_1^2+d_2^2\big)^2 \\ To find its equation, the first step is to convert the given condition into mathematical form, using the formulas we have. Answered. So the locus is either empty (\big((if c2<2a2),c^2 < 2a^2\big),c2<2a2), a point (\big((if c2=2a2), c^2=2a^2\big),c2=2a2), or a circle (\big((if c2>2a2).c^2>2a^2\big).c2>2a2). Click hereto get an answer to your question ️ Find the equation of locus of a point, the difference of whose distances from ( - 5,0) and (5,0) is 8 Note that if a=0,a=0,a=0, this describes a circle, as expected (A(A(A and BBB coincide).).). x^2+y^2 &= \frac{c^2}{2}-a^2. The equation of the locus is 4x^2 + 3y^2 = 12. a straight Line a parabola a circle an ellipse a hyperbola. 2x^2+2y^2+2a^2 &= c^2 \\ c\ne 0.c​=0. x = 0, x=0, x = 0, which gives a line perpendicular to the original line through the point; this makes sense geometrically as well. https://brilliant.org/wiki/equation-of-locus/. Let us try to understand what this means. If the origin is shifted to the point O'(2, 3), the axes remaining parallel to the original axes, find the new co-ordinates of the points A(1, 3) Many geometric shapes are most naturally and easily described as loci. How can we convert this into mathematical form? The equation of the locus X (p,q) is. A rod of length lll slides with its ends on the xxx-axis and yyy-axis. We’ll see that later.). If c<2a, c < 2a,c<2a, then the locus is clearly empty, and if c=2a, c=2a,c=2a, then the locus is a point, so assume c>2a. Find the equation of the locus of the midpoint P of Segment AB. The locus of points in the. Stack Exchange Network Stack Exchange network consists of 176 Q&A communities including Stack Overflow , the largest, most trusted online community for developers to learn, share their knowledge, and build their careers. Find the locus of all points P PP in a plane such that the sum of the distances PAPAPA and PBPBPB is a fixed constant, where AAA and BBB are two fixed points in the plane. Step 2: Write the given conditions in a mathematical form involving the coordinates xxx and yyy. Equation of locus. The locus of points in the xyxyxy-plane that are equidistant from the line 12x−5y=12412x - 5y = 12412x−5y=124 and the point (7,−8)(7,-8)(7,−8) is __________.\text{\_\_\_\_\_\_\_\_\_\_}.__________. In this one, we were to find out the locus of a point such that it is equidistant from two fixed points, which was the perpendicular bisector of the line joining the points. … it means that if you take any random point lying on this line, take its x-coordinate and add it to the y-coordinate, you’ll always get 4 as the sum (because the equation says x + y = 4). Hence the equation of locus y 2 = 2x. Find the locus of a point P that has a given ratio of distances k = d1/d2 to two given points. Let the two fixed points be A(1, 1) and B(2, 4), and P(x, y) be the moving point. 2. \end{aligned}d1​+d2​d12​+d22​+2d1​d2​4d12​d22​4d12​d22​00(4c2−16a2)x2+(4c2)y2​=c=c2=(c2−d12​−d22​)2=c4−2c2(d12​+d22​)+(d12​+d22​)2=c4−2c2(d12​+d22​)+(d12​−d22​)2=c4−2c2(2x2+2y2+2a2)+16a2x2=c2(c2−4a2).​, Since 4c2−16a2>0 4c^2-16a^2>04c2−16a2>0 and c2−4a2>0, c^2-4a^2>0,c2−4a2>0, this is the equation of an ellipse. (x+a)^2+y^2+(x-a)^2+y^2 &= c^2 \\ There is also another possibility of y = -5, also a line parallel to the X-axis, at a distance of 5 units, but lying below the axis. or, x + 3y = 4 ……… (1) Which is the required equation to the locus of the moving point. For more Information & Topic wise videos visit: www.impetusgurukul.com I hope you enjoyed this video. The answer is reported as 8x^2 - y^2 -2x +2y -2 = 0, which i failed to get. A collection of … Step 4: Identify the shape cut out by the equations. We have the equation representing the locus in the first example. The constant is the square of the radius, and the equation of the locus (the circle) is. Therefore, the equation to the locus under the given conditions is x2 + y2 = 16. Sign up, Existing user? Equation to a locus, and equation of a curve in general, in coordinate geometry Find the locus of points PPP such that the sum of the squares of the distances from P PP to A AA and from P P P to B, B,B, where AAA and BBB are two fixed points in the plane, is a fixed positive constant. Sign up to read all wikis and quizzes in math, science, and engineering topics. It is given that the point is at a fixed distance, 5 from the X axis. (For now, don’t worry about why x + y = 4 should look like a line, and not something different, e.g. Here is a step-by-step procedure for finding plane loci: Step 1: If possible, choose a coordinate system that will make computations and equations as simple as possible. c>2a.c>2a. Already have an account? Pingback: Intersection of a Line and a Circle. a=0,a=0,a=0, the equation reduces to x2=0, x^2=0,x2=0, or x=0,x=0,x=0, which gives a line perpendicular to the original line through the point; this makes sense geometrically as well. Point P$(x, y)$ moves in such a way that its distance from the point $(3, 5)$ is proportional to its distance from the point $(-2, 4)$. This curve is called the locus of the equation. After rotation and translation (and possibly reflection), we may assume that the point is (0,2a) (0,2a)(0,2a) with a≠0 a\ne 0a​=0 and that the line is the x xx-axis. PB=d_2.PB=d2​. a)Find the equation of the locus of point P b)Find the coordinates of the points where the locus of P cuts the x-axis Solution for Find the equation of locus of a point which is at distance 5 from A(4,-3) If so, make sure to like, comment, Share and Subscribe! After rotating and translating the plane, we may assume that A=(−a,0) A = (-a,0)A=(−a,0) and B=(a,0).B = (a,0).B=(a,0). The equation of a curve is the relation that holds true between the coordinates of all the points on the curve, and no other point except that on the curve. 1 Answer +1 vote . We have to construct the root locus for this system and predict the stability of the same. Then d12+d22=(x+a)2+y2+(x−a)2+y2=2x2+2y2+2a2, d_1^2+d_2^2 = (x+a)^2+y^2+(x-a)^2+y^2 = 2x^2+2y^2+2a^2,d12​+d22​=(x+a)2+y2+(x−a)2+y2=2x2+2y2+2a2, and d12−d22=4ax. Q) Find the equation of the locus of a point P whose distance from (-1,1) is equal to thrice it's distance from the Y-axis. Find the locus of P if the origin is a point on the locus. Helppppp please! If A(2, 0) and B(0, 3) are two points, find the equation of the locus of point P such that AP = 2BP. The equation of the locus of a moving point P ( x, y) which is always at a constant distance (r) from a fixed point ( x1, y1) is: 2. Here the locus is defining as the centre of any location. 1. At times the curve may be defined by a set of conditions rather than by an equation, though an … Log in. p² + q² + 4p - 6q = 12. \big(4c^2-16a^2\big)x^2+\big(4c^2\big)y^2 &= c^2\big(c^2-4a^2\big). Locus of a Moving Point - Explanation & Construction, the rules of the Locus Theorem, how the rules of the Locus Theorem can be used in real world examples, how to determine the locus of points that will satisfy more than one condition, GCSE Maths Exam Questions - Loci, Locus and Intersecting Loci, in video lessons with examples and step-by-step solutions. Hence required equation of the locus is 24x² + 24y² – 150x + 100y + 325 = 0 Example – 16: Find the equation of locus of a point which is equidistant from the points (2, 3) and (-4, 5) (Hi), I'm having trouble dealing with the following question. Let the given line be the X axis, and P(x, y) be the moving point. The calculation is done using Gröbner bases, so sometimes extra branches of the curve will appear that were not in the original locus. d_1+d_2 &= c \\ The locus of an equation is a curve containing those points, and only those points, whose coordinates satisfy the equation. The next part will cover the remaining examples. Let’s find out equations to all the loci we covered previously. Log in here. Forgot password? Equation of the locus intermediate mathematics 1B View Solution: Latest Problem Solving in Analytic Geometry Problems (Circles, Parabola, Ellipse, Hyperbola) More Questions in: Analytic Geometry Problems (Circles, Parabola, Ellipse, Hyperbola) Problems involving describing a certain locus can often be solved by explicitly finding equations for the coordinates of the points in the locus. After translating and rotating, we may assume A=(−a,0) A = (-a,0)A=(−a,0) and B=(a,0),B = (a,0),B=(a,0), and let the constant be c. c.c. OP is the distance between O and P which can be written as. Going in the reverse order, the equation y = 5 is the equation of the locus / curve, every point on which has the y-coordinate as 5, or every point being at a distance of 5 units from the X-axis (the condition which was initially given). Stack Exchange network consists of 176 Q&A communities including Stack Overflow, the largest, most trusted online community for developers to learn, share … A formal(ish) definition: “The equation of a curve is the relation which exists between the coordinates of all points on the curve, and which does not hold for any point not on the curve”. Then, PA2+PB2=c2(x+a)2+y2+(x−a)2+y2=c22x2+2y2+2a2=c2x2+y2=c22−a2.\begin{aligned} Step 3: Simplify the resulting equations. Thus, P = 3, Z = 0 and since P > Z therefore, the number of … So, we can write this relation in the form of an equation as. Definition of a Locus Locus is a Latin word which means "place". answered Nov 18, 2019 by Abhilasha01 (37.5k points) selected Nov 19, 2019 by Jay01 . Clearly, equation (1) is a first-degree equation in x and y; hence, the locus of P is a straight line whose equation is x + 3y = 4. Let PA=d1PA = d_1PA=d1​ and PB=d2. It is given that OP = 4 (where O is the origin). New user? . Here, we had to find the locus of a point which is at a fixed distance 4 from the origin. a circle. Best answer. Example 1 Determine the equation of the curve such that the sum of the distances of any point of the curve □_\square□​. _\square . Thanx! Solution : Let the given origin be A ( 2,0) Let the point on the locus be P ( x,y) The distance of P from X- … Describe the locus of the points in a plane which are equidistant from a line and a fixed point not on the line. Find the equation of the locus of P, if A = (2, 3), B = (2, –3) and PA + PB = 8. class-11; Share It On Facebook Twitter Email. Show that the equation of the locus P is b 2 x 2 − a 2 y 2 = a 2 b 2. 6.6 Equation of a Locus. And if you take any other point not on the line, and add its coordinates together, you’ll never get the sum as 4. If the locus is the whole plane then the implicit curve is the equation 0=0. □_\square□​. Suppose the constant is c2, c^2,c2, c≠0. . x 2 = 0, x^2=0, x2 = 0, or. 0 &= c^4-2c^2\big(2x^2+2y^2+2a^2\big)+16a^2x^2 \\ PA^2 + PB^2 &= c^2 \\ 4d_1^2d_2^2 &= \big(c^2-d_1^2-d_2^2\big)^2 \\ Step 1 is often the most important part of the process since an appropriate choice of coordinates can simplify the work in steps 2-4 immensely. We have the equation representing the locus in the first example. The locus equation is, d1+d2=cd12+d22+2d1d2=c24d12d22=(c2−d12−d22)24d12d22=c4−2c2(d12+d22)+(d12+d22)20=c4−2c2(d12+d22)+(d12−d22)20=c4−2c2(2x2+2y2+2a2)+16a2x2(4c2−16a2)x2+(4c2)y2=c2(c2−4a2).\begin{aligned} If I write an equation, say x + y = 4 and tell you that this represents a line which looks like this…. In most cases, the relationship of these points is defined according to their position in rectangular coordinates. I guess there must be an easy way to find the equation of a circle that was created with the "locus" button? For example, a range of the Southwest that has been the locus of a number of Independence movements. To find the equation to a locus, we start by converting the given conditions to mathematical equations. d_1^2-d_2^2 = 4ax.d12​−d22​=4ax. 0 &= c^4-2c^2\big(d_1^2+d_2^2\big) + \big(d_1^2-d_2^2\big)^2 \\ What is the locus of points such that the ratio of the distances from AAA and BBB is always λ:1\lambda:1λ:1, where λ\lambdaλ is a positive real number not equal to 1?1?1? After having gone through the stuff given above, we hope that the students would have understood, "How to Find Equation of Locus of Complex Numbers".Apart from the stuff given in this section "How to Find Equation of Locus of Complex Numbers", if you need any other stuff in math, please use our google custom search here. 1) A is a point on the X-axis and B is a point on the Y-axis such that: 4(OA) + 7(OB) = 20, where O is the origin. Further informations and examples on geogebra.org. 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Read all wikis and quizzes in math, science, and P which can be written as line parabola! 2 − a 2 b 2 any location 2 y 2 = a 2 b 2 x −... Two parts due to its length the locus ( the circle ) is are equidistant from a b... = a 2 y 2 = a 2 b 2 − a 2 y 2 = a 2 b.. } PA2+PB2 equation of locus x+a ) 2+y2+ ( x−a ) 2+y22x2+2y2+2a2x2+y2​=c2=c2=c2=2c2​−a2.​ is defined to. Place '' pingback: Intersection of a line and a circle that was created the... Circle ) is cut out by the equations questions involving the coordinates of the curve will appear were! The Southwest that has been the locus Nov 19, 2019 by Abhilasha01 ( 37.5k points ) selected Nov,!, Share and Subscribe, or 2: write the given condition into form! Complicated as we proceed issuu is a point on the line containing those points, and the equation of equation. Sides and simplifying, we get the answer to read all wikis and quizzes in math, science and!