Kindly solve all of these questions from Question Number 1 to Question Number 15 . Do not leave any question . Solve all of them .
1. Find the ratio in which the point (2. y) divides the line segment joining (4, 3) and (6,3) and hence 2 Find the locus of the middle point of the portion of the line x cosu y sina wci 3. If the vertices of a triangle have integral coordinates, prove that the trisngle cannot be 4. Two consecutive sides of a paralielogram are 4x 5y 0 and 7x+2y 0 If the equation of one find the value of y intercepted between the axes, given that \'p\' remains constant equilateral of the diagonals is 11x+ 7y 9, find the equation of the other The equation of the base of an equilateral triangle is x + 2 and the vertex is (2-1). Find the diagonal. length of the side of the triangle. 5/ Find if (, 2) is an interior point of AABC formed by x + y = 4, 3x-7y-8, 4x-y=31 7A variable straight line passes through the points of intersection of the lines x 2y section of the lines x + 2y = 1 and meets the co-ordinates axes in A and B. Prove that the locus of the midpoint of 2x -y 1 and As is 10xy = x + 3y. A right angled triangle ABC having a right angle at C, CA b and CB a, move such that the A right angled triangle ABC having a right angle at C, CA-b and CB angular points A and B slide along x-axis and y-axis respectively. Find the locus of C Find the equation of straight lines passing through-2-7 and having an intercept of length units between the straight lines 4x+3y 12, 4x3y 3 Let ABC be a triangle having orthocentre and circumcentre at (9, 5) and (0, 0) respectively. If the 10. tion of side BC is 2x-y= 10, then find the possible coordinates of vertex A. The base of a triangle is axis of x and its other two sides are given by the equations y=-x+(1+) and y=1+px+(1+p) Prove that locus of its orthocentre is the line x + y 0 12. A line intersects x-axis at A(7, 0) and y-axis at B(0. 5). A varlable line PO which is perpendicular to AB intersects x-axis at P and y-axis at Q. If AQ and BP intersect at R, then find the locus of R. Lines L., ax + by + c-oand L2 = 1x + my + n-ointersect at the point P and makes an angle @ 13. with each other. Find the equation of a line L different from La which passes through P and makes the same angle with L1. 14. The straight line1 cuts the axes in A and B and a line perpendicular to AB oults the axes in P and Q. Find the locus of the point of intersection of AQ and BP A variable line L passing through the point B (2, 5) intersects the lines 2x -5xy 2y\' 0 at P and QFind the locus of the point R on L such that distances BP, BR and BQ are in harmonic and Q progression. 15. A
Hi, since 1 question is to be posted per page, please post Question 2 and onwards separately. I am taking up Q1)
Answer 1)
Let 2,y divide the line joining (4,3), (6,3). in m:n ratio. Hence,
m*6-n*4/ m-n , m*3-n*3/ m-n is the point,
Take y, which is m*3-n*3/ m-n = 3(m-n)/(m-n) = 3. Hence, y=3.
This could have been calculted mentally , as we know that y values for given points are 3. Hence the line which passes through both the point will have a y value of 3. Hence the point 2,y is 2,3
2=(m*6-n*4)/( m-n) , 2m-2n=6m-4n
2n=4m, m/n = 1/2
Hence the ratio in which 2,3 divides the line segment is 1:2
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