Follow the below given steps to find the equation of locus of a point:<\/p>\n
Step I<\/b> Assume the coordinates of the point say (h,k) whose locus is to be found.<\/p>\n
Step II<\/b> Write the given condition in mathematical form involving h, k.<\/p>\n
Step III<\/b> Eliminate the variable(s), if any.<\/p>\n
Step IV<\/b> Replace h by x and k by y in the result obtained in step III. The equation so obtained is the locus of the point, which moves under some stated condition(s).<\/p>\n
STRAIGHT LINE<\/span><\/b>is a locus of the point P (h,k)<\/i> which moves in such a way that point P is collinear with the two given points. We can also define straight line as a curve such that the every point on the line segment joining any two points on it lies on it. Line is a 1-D curve and thus its equation will be of degree one.<\/p>\n \u00c3\u201a\u00c2\u00a0If we consider a single point, definitely infinite numbers of straight lines can be drawn passing through it. But if we are given two points, then one and only one line passes through it. Thus we require at least two conditions to get one unique line. Depending upon the two conditions available with us, we can write the equations in different forms. Some of them are:<\/span><\/p>\n A.\u00c3\u201a\u00c2\u00a0 \u00c3\u201a\u00c2\u00a0<\/span><\/span><\/b>TWO-POINT FORM OF A LINE:<\/span><\/b><\/p>\n Consider two points A(x1<\/sub>, y1<\/sub>) and B(x2<\/sub>, y2<\/sub>). Let C(x,y) be any point which is collinear with A and B (condition for any point to be on a line).\u00c3\u201a\u00c2\u00a0<\/span>Thus we can say, slope of AB = slope of AC<\/span><\/p>\n C. SLOPE-INTERCEPT FORM OF A LINE<\/b>:<\/p>\n Now consider the equation obtained in (2) and assume that this line passes through (0,c) on the y – axis.<\/p>\n Thus we can replace (x1<\/sub>, y1<\/sub>) by (0,c). The equation will become<\/p>\n y-c = mx<\/p>\n or y = mx + c<\/p>\n where c is called the y intercept of the line.<\/p>\n This is the required equation of the line in slope- intercept form.<\/p>\n D. INTERCEPT FORM OF A LINE:<\/b><\/p>\n In this form of equation of line, we will be given with the x and y intercept. So we can easily get the co-ordinates of the points of line on x and y axis respectively.<\/p>\n If x-intercept is a, that means line passes through (a,0) on the x-axis. Similarly, if y-intercept is b, that means line passes through (0,b) on the y \u00c3\u00a2\u00e2\u201a\u00ac\u00e2\u20ac\u0153 axis. Using these two points, we can easily write the equation of line as,<\/p>\n E. NORMAL FORM OR PERPENDICULAR FORM OF A LINE:<\/b><\/p>\n In this form of the straight line, the length of perpendicular from the origin and an angle which this perpendicular makes with positive direction of x-axis is known to us.<\/p>\n Let AB be the line such that the length of the perpendicular OQ from the origin O to the line be p and \u00c3\u00a2\u00cb\u2020\u00c2\u00a0 XOQ = \u00c3\u017d\u00c2\u00b1.<\/p>\n From the diagram, we can easily find the length OA and OB using trigonometric ratios.<\/p>\n Thus OA = p sec \u00c3\u017d\u00c2\u00b1 and OB = p cosec \u00c3\u017d\u00c2\u00b1<\/p>\n These OA and OB are actually the x and y intercepts of the line AB.<\/p>\n Thus using intercept form, we can write,<\/p>\n In this form of equation of a line, a point and its inclination with the positive direction of x \u00c3\u00a2\u00e2\u201a\u00ac\u00e2\u20ac\u0153axis (in anticlockwise direction) is known to us. By knowing the inclination angle, we can also find the slope but sometimes writing the equation in terms of \u00c3\u00a2\u00e2\u201a\u00ac\u00cb\u0153theta\u00c3\u00a2\u00e2\u201a\u00ac\u00e2\u201e\u00a2 as a new parameter simplifies the given problem. Therefore we go for this form of a line.<\/p>\n Let\u00c3\u00a2\u00e2\u201a\u00ac\u00e2\u201e\u00a2s now analysis the situation and derive the equation of line in parametric form. Again, if consider a point is taken on the line below the given point A (x1<\/sub>, y1<\/sub>), then from the \u00c3\u00a2\u00cb\u2020\u00e2\u20ac\u00a0APN\u00c3\u00a2\u00e2\u201a\u00ac\u00e2\u201e\u00a2<\/p>\n Any line parallel to a given line will have the same slope but different intercept. So the equation of a line parallel to a given line ax + by + c = 0 is Any line parallel to a given line will have the same slope but different intercept. So the equation of a line parallel to a given line ax + by + c = 0 is\u00c3\u201a\u00c2\u00a0\u00c3\u201a\u00c2\u00a0 \u00c3\u201a\u00c2\u00a0 \u00c3\u201a\u00c2\u00a0 \u00c3\u201a\u00c2\u00a0 \u00c3\u201a\u00c2\u00a0 \u00c3\u201a\u00c2\u00a0 \u00c3\u201a\u00c2\u00a0 \u00c3\u201a\u00c2\u00a0 \u00c3\u201a\u00c2\u00a0 \u00c3\u201a\u00c2\u00a0 \u00c3\u201a\u00c2\u00a0 \u00c3\u201a\u00c2\u00a0 \u00c3\u201a\u00c2\u00a0 \u00c3\u201a\u00c2\u00a0Where \u00c3\u017d\u00c2\u00bb is a constant and value of \u00c3\u017d\u00c2\u00bb can be determined using another given condition.<\/p>\n In this section we will discuss about the combined equation of two lines. When any two line\u00c3\u201a\u00c2\u00a0Intersects at a point, than we can write\u00c3\u201a\u00c2\u00a0 a single equation representing the two lines. The angle \u00c3\u017d\u00c2\u00b8 between the two lines represented by a general equation is the same as that between the two lines represented by its homogeneous part only.<\/p>\n Pair of straight lines through origin means origin is the common point of intersection and thus point O (0, 0) will satisfy any general equation of pair of straight line.<\/p>\n Note:<\/b>\u00c3\u201a\u00c2\u00a0 A homogeneous equation of degree n represents n straight lines through the origin.<\/p>\n I am Vijay Mukati<\/strong><\/a>, Mathematics Expert at askIITians. I did my post-graduation from IIT Patna. For more about me you can visit the following link<\/p>\n You can also refer to some of my Video Lectures in Physics, available on the following links.<\/p>\n\n
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<\/a><\/p>\n<\/a><\/p>\n<\/a><\/p>\n<\/a><\/p>\n<\/a><\/p>\n<\/a>F. \u00c3\u201a\u00c2\u00a0<\/b>PARAMETRIC FORM OF A LINE:<\/b><\/p>\n
\nConsider a point P (x, y) taken on the line above the given point (x1<\/sub>, y1<\/sub>)<\/p>\n<\/a><\/p>\n<\/a><\/p>\n<\/a><\/p>\n\n
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<\/a><\/p>\n<\/a><\/p>\n<\/a><\/p>\n\n
\nWhere \u00c3\u017d\u00c2\u00bb is a constant and value of \u00c3\u017d\u00c2\u00bb can be determined using another given condition.<\/p>\n\n
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<\/a><\/p>\n<\/a><\/p>\n<\/a><\/p>\n