Okay, friends, today we are going to discuss about some miscellaneous problem on straight lines. We have already discussed various concept on a straight line. Now we shall discuss some miscellaneous problem, so first problem is find the equation of line which passes through the point minus 4 3, and the portion of the line intercepted between the axis is divided internally in the ratio. Five is to three by the this point, so here we have figure of this problem. We have to find the equation of this line, a b which passing through this point, p minus four three, and this point p divides this a b in the ratio. Five is to three so first of all, we find the point b and a by using section formula. We know that if this point p divide this a b in internally then minus four equal to 3 into a plus 5 into 0 by 5 plus three. So this is three a by eight, so this implies three a is equal to minus 32. This implies a is equal to minus 32 by 3. This 3 is equal to 5 into b plus 3 into 0 by 5 plus 3. So this implies five b equal to twenty four. This implies b is equal to twenty four by five we get the value of, or we get the coordinate of this two point: a and b, so a is equal to coordinate of a is equal to minus 32 by 3, 0 and b is equal to 0 24 by 5., so equation of line equation of line a b x by minus thirty, two by three plus y by twenty four by five equal to one. So this implies three x by minus thirty, two plus five y by twenty four equal to one. This implies now take lcm or we can write it as three x by four minus five y by three is equal to minus eight, so this implies nine x minus twenty y by twelve equal to minus 8. This implies 9 x minus 20 y equal to minus 96. This implies 9 x minus twenty y plus. Ninety six equal to zero is a required equation of line. So in this way we can find any line which passing through or which make intercept, x, intercept and y intercept, and this line having some point given, then we can use section formula and find the equation of line. Now we have another problem that is find the values of k for which the line k minus three x, minus four minus k, square y plus k square minus seven k, plus six equal to zero, is first parallel to x axis. Second, parallel, two y axis and third passes through origin when line parallel to x axis so parallel to x. Axis means what the coefficient of x equal to zero. So parallel to x, axis first case, parallel to x axis, implies coefficient of x equal to 0. This implies k minus 3 equal to 0. This implies k is equal to now. Question is y coefficient of x equal to 0, because when line parallel to x axis then is slope equal to zero thats. Why coefficient of x equal to zero? Now? Second, parallel to y axis parallel to y axis, what does it mean parallel to axis means slope of that line equal to infinity, or you can say, not defined. It means the coefficient of y must be equal to zero, so this implies coefficient of y equal to zero, so coefficient of y equal to zero means four minus k square equal to zero. This implies four minus k square equal to zero. This implies k square equal to four, so this implies k is equal to plus minus two third passes through origin passes through, or is it so any line passes through origin? That line must be like this means its c equal to 0 means when you take y, is equal to m x, plus c. So if line passes through origin, so then c must be equal to zero. It means so c. Equal to 0 implies k square minus 7 k plus 6 equal to 0. This implies k square minus six k, minus k plus six equal to zero, so this implies k minus 6 equal to 0. So this imply k minus 1 to k minus 6 equal to zero. This implies k equal to one or six, so for these two value line is passing through origin. So in this way we can discuss various condition. Now we have another problem that is find the equation of one of the sides of an isoscele isosceles right triangle: triangle which hypotenuse is given by three x, plus four y four and the opposite. Vertex of the hypotenuse is two two: we have to find equation of one of this side, so what is given? We have. We have given equation of hypotenuse and its coordinate of opposite vertices, and the side makes 45 degree because right as well as right angle. So this is 45 degree: slope of a b slope of a b equal to minus three by four slope of a b equal to minus three by four: let slope of ac equal to m. Now this line ac and a b max angle 45 degree. So ten forty five degree is equal to mod m minus m one 1 plus m m 1. So this implies 1 is equal to m minus minus three by four by one plus m minus three by four. So we can write it as one is equal to 4 m minus plus 3 by 4 by four minus three m by four cancel, so one is equal to 4m plus 3 and 4 minus 3m. So this implies 4m plus 3 by four minus three m is equal to plus minus one. So this implies four m plus three is equal to four minus three m or four m plus three is equal to minus four plus three m, so this implies. Seven m. Is equal to one or m is equal to minus seven, so this implies m is equal to one by seven or m is equal to minus seven. We have two equation which passing through this point c, two two so equation of line with slope m equal to one by seven y minus 2, equal to 1 by 7 x, minus 2. So this implies 7 y minus 14 is equal to x, minus two, so x, minus seven y plus twelve equal to zero, again equation of line with slope m equal to minus seven. So y minus 2 is equal to minus 7 x minus 2. This implies y minus 2 is equal to minus seven x, plus fourteen, so seven x, plus y and minus sixteen equal to zero.So these are the two equation of line. That is, we have a c and a b we have if one diagonal of a square is along the line. Eight x, minus fifteen y equal to zero and one of its vertex is at one. Two then find the equation of sides of square passing this vertex. So this problem is just like this problem. We have discussed so here this line this. This is hypotenuse with opposite vertex a is given, so we can find. Let us solve this problem, so slope of diagonal equal to eight by fifteen and let the slope of side a b equal to m, so ten forty five degree why this is ten forty five degree, because you know that each angle of square is ninety degree and Diagonal is angle bisector, so this is 45 degree. So 1045 degree is equal to again mod m minus m 1 by 1 plus mm 1 say this is m1, so this implies 1 is equal to m minus eight by fifteen. One plus m into eight by fifteen whole mod, so this implies one is equal to fifteen m minus eight by fifteen plus item. So fifteen m, minus eight by fifteen plus eight m, is equal to plus minus 1. So this implies 15 m. Minus 8 is equal to 15 plus 8 m r. 15 m. Minus 8 is equal to minus 15 minus 8 m, so this implies 7 m is equal to 23 or twenty three m is equal to minus seven. So m is equal to twenty three by seven or m is equal to minus seven by twenty three equation of side of a square with slope m is equal to twenty three by 7 and passing through a one. Two y minus two is equal to twenty three by seven x minus 1, so this implies 7 y minus 14 is equal to 23 x minus 23. So twenty three x minus seven y and minus nine equal to zero, again equation of side with slope m equal to minus seven by twenty three and passing through a one. Two is y minus two equal to minus seven by twenty three and x minus one. This implies 23y minus 46 equal to minus 7x plus 7.. This implies 7 x plus 23 y minus 53 equal to 0. So in this way we can find the equation of side a b and a d now another problem - and this is most interesting problem - find the image of point three: eight, with respect to line x, plus three y equal to seven, assuming the line to be a Plane mirror equation of line x, plus three y equal to seven this line. We assume this line as a mirror, and here this point is given p equal to three eight. We have to find the image of this point. P say the image of this point. P. 3 8 is q, alpha beta and let this p q intersect. This line at n and this point n is called foot of perpendicular. So we can solve this problem in various way. But let us try to find the this point. N. So slope of l, slope of l means given line is minus one by three. So slope of p q, which is perpendicular to l equal to minus minus one by three, is equal to three, because we know that the slope of perpendicular line related like m one into m two equal to minus one, so slope of perpendicular line is negative. Reciprocal now we have two information for this line. P q slope is three and passing through three eight, so equation of line p q with slope three and passing through p. Three: eight y minus eight is equal to three x minus three, so three x, minus y and minus one equal to zero three x, minus y minus one equal to zero. So say this is first and given equation of line x, plus three y equal to seven. So this implies x is equal to seven minus three y put x equal to seven minus three y in one, so three into seven minus three y minus y minus one equal to zero. So this implies twenty one: minus nine y minus y minus one equal to zero, so minus ten y plus twenty equal to zero. This implies y is equal to two. So x is equal to 7 minus 3 y 7. Minus 3 into 2 means seven minus six. Equal to one, so the coordinate of foot of perpendicular n is one two, so the coordinate of this foot of perpendicular is one two. Now this n is midpoint of this p q, this any midpoint of p q. It means this is p, and this is q. Q is alpha beta and p is 3 8, and this point n is 1 2, which is midpoint of p q. Since n is the midpoint midpoint of p q, so alpha plus three by two is equal to one. This implies alpha plus three is equal to two. This implies. Alpha is equal to minus one, and beta plus eight by two is equal to two. This implies beta plus eight is equal to four, so this implies beta equal to minus four. So image of point point p: three: eight, with respect to a line x, plus three y equal to seven - is q minus one minus four in another way. We can also find this image of point say this p q. The this line intersect at midpoint, so find the value of this n in terms of alpha and beta and put that value in this equation. You will get the value of alpha and beta now another problem. If the area of the triangle formed by a line with coordinate axis is 54 root, 3 square unit and the perpendicular drawn from the origin to the line makes an angle of 60 degree with x axis find the equation of the line. We have to find the equation of this line. A b this a b may intercept with x axis and y axis say this is a and b and this line form an angle say this is origin, o a b and the area of this triangle, o a b is given as fifty four root. Three given area of triangle, o a b, is equal to fifty four root. Three, let the equation of line be x by a plus y, by b equal to one. So the coordinate of this a is a zero and coordinate of this b is zero b. Now we have two information: one formation is given that perpendicular from origin max angle, 60 degree with the x axis. It means alpha is given, so we can use this line equation of line in normal form x. Cos alpha, plus y sine alpha equal to p p. Is length of perpendicular from origin, or you can say distance of this line a b from origin, so this implies x, cos 60 degree plus y sign sixty degree equal to p. This implies x into 1 by 2 plus y into root 3 by 2, equal to p. This implies x by two p plus y by two p, by root three equal to one. This is second so compare on comparing one and two will have a equal to two p and b equal to two p by root. Three given area of triangle, o a b equal to fifty four root, three square unit, so for right hand, angle triangle: you know that half into base into height, so half into a into b equal to fifty four root three, so this implies half into 2 p Into 2 p by root 3 is equal to 52 root 3. This implies four p. Two p squared is equal to 54 into 3. This implies p square equal to 81. This implies p is equal to plus minus 9 plus minus 9 question. Is we have to find the equation of line so now for this equation? We have length of perpendicular is known, and this alpha is known so equation of line equation, of line with alpha equal to 60 degree and p is equal to plus minus 3 is x, cos alpha, plus y sine alpha equal to p. This implies x, cos 60 degree plus y sine 60 degree equal to 3 or x, cos 60 degree plus y sine sixty degree equal to minus three. So this implies x into one by two plus y into root: 3 by 2, equal to 3 and x into 1 by 2 plus y into root 3 by 2, equal to minus 3, so x, plus root 3y minus 6, equal to 0 or x plus root. Three y plus six equal to zero. Another problem that is find the distance of the point p four one from the line four x minus y equal to zero, measured along the line, making an angle of one thirty, five degree with the positive direction of x axis. What is given equation of this line is given. This l is given four x minus y equal to zero, and we have to find the distance of this line. From this point p. Four one situated on this line say this line is l, one which makes angle one thirty five degree in the positive direction of x, axis slope of given equation of line given equation of line four x minus y equal to zero. This is line. L say this is equation. One also given line l one max angle, one thirty five degree with x axis in positive direction, so slope of l one is equal to ten one. Thirty, five degree: it means minus one. Now for this line, we have two information. That is, its slope is minus one, and this line passing through point p. Four one so equation of line. L, one with slope m equal to minus 1 and passing through p 4 1 is y minus 1 equal to minus 1 x minus four. This implies y minus one equal to minus x, plus four, so x, plus y minus five equal to zero say this is line two and line. One is what four x minus y equal to zero is line one. So this implies y equal to four x. So, put y equal to four x into so x, plus 4 x, minus 5 equal to 0, so this implies 5 x equal to 5. So this implies x equal to one so y equal to four. It means we get the point of inverse this point: q. As one four now question is find the distance of point p, four, one from the line four x minus y equal to zero, measured along the line, making an angle one thirty five degree with the positive direction of x axisement means we have to find the distance Of this line from along with this line, so we just find the distance between this p q, so distance of line l from p 4 1 along the line. L one is p q, l one is p q, so p q equal to four minus one whole square plus one minus four whole square. It means three square plus 3 s square, so 3 root 2 unit. So distance of this line from this point along this linear one - is there two units now so if the line three x plus y minus two equal to zero p x, plus two, i minus three equal to zero and two x minus y minus three equal to Zero are concurrent find the value of p, since we have three equation of line is given, and these three are lines are concurrent concurrent means these three lines passing through one point, so different line three or more than three lines passing through same point, is called concurrent Length this c line passing through this point. Then we have to find the value of so if any three line say a one x, plus b, one y plus c one equal to zero. A two x plus b, two y plus c, two equal to zero. A three x plus b, three y plus c three equal to zero. If these three lines are concurrent, if these c lines are concurrent, then a 1 b, 1 c 1, a 2 b, 2 c, two, a three b, three c three equal to zero. We just try to expand this quantity known as determinant when we have to find the value of this determinant. We expand and we follow this sign rule so a 1 a 1 then b, 2 c 3, minus b, 3 c, 2 b, 2 c 3, minus b, 3 c 2, minus b, 1 and b 1 means a 2 c, 3 minus a 3 c 2. A 2 c three minus a three c, two plus c one, so a two b, three minus a three b, two equal to zero, now come to the problem. We have given these three equation of line. So, given equation of line lines are three x plus y minus two equal to zero p x, plus two, i minus three equal to zero and two x minus y minus three equal to zero. So given since say this is 1 2 and say this is three since given lines, one two and three are concurrent concurrent, so 3, 1 minus 2, p, 2, p, 2, minus 3, 2 minus 1 minus 3 equal to 0. This implies 3 2 into minus 3. Minus 6 - and this is minus 3, say plus minus plus so minus 1, so minus 3 p - and this is plus 6 - and this is minus 2 minus p minus four equal to zero. So this implies minus nine minus twenty seven, and this is plus three p. Minus six plus two p, plus eight equal to zero. This implies five p and this is minus thirty, three plus eight equal to zero minus thirty, three plus eight equal to zero. So this is five p minus twenty five equal to zero, so p equal to five. So in this way we can find the value of unknown quantity if equation of lines are concurrent by using this condition. So, ok, we will discuss some other problems in next session. Thank you. You