There are three coplanar parallel lines. If any `p` points are taken on each of the lines, the maximum number of triangles with vertices on these points is a. `3p^2(p-1)+1` b. `3p^2(p-1)` c. `p^2(4p-3)` d. none of these

P is a point on the line y+2x=1, and Q and R two points on the line 3y+6x=6 such that triangle P Q R is an equilateral triangle. The length of the side of the triangle is (a) 2/(sqrt(5)) (b) 3/(sqrt(5)) (c) 4/(sqrt(5)) (d) none of these

If A(1,p^2),B(0,1) and C(p ,0) are the coordinates of three points, then the value of p for which the area of triangle A B C is the minimum is 1/(sqrt(3)) (b) -1/(sqrt(3)) 1/(sqrt(2)) (d) none of these

If P is a point (x ,y) on the line y=-3x such that P and the point (3, 4) are on the opposite sides of the line 3x-4y=8, then

Find the co-ordinates of point P if P divides the line segment joining the points A(-1,7) and B(4,-3) in the ratio 2:3

If the roots of the quadratic equation (4p-p^2-5)x^2-(2p-1)x+3p=0 lie on either side of unit, then the number of integer values of p is a. 1 b. 2 c. 3 d. 4

For the three events A ,B ,a n dC ,P (exactly one of the events AorB occurs) =P (exactly one of the two evens BorC ) =P (exactly one of the events CorA occurs) =pa n dP (all the three events occur simultaneously) =p^2w h e r e 0 < p < 1//2. Then the probability of at least one of the three events A ,Ba n dC occurring is a. (3p+2p^2)/2 b. (p+3p^2)/4 c. (p+3p^2)/2 d. (3p+2p^2)/4

If C_1: x^2+y^2=(3+2sqrt(2))^2 is a circle and P A and P B are a pair of tangents on C_1, where P is any point on the director circle of C_1, then the radius of the smallest circle which touches c_1 externally and also the two tangents P A and P B is 2sqrt(3)-3 (b) 2sqrt(2)-1 2sqrt(2)-1 (d) 1

A point p moves such that p and the points (2,3)(1,5) are always collinear. Show that the locus of p is 2x+y-7=0

Find the coordinates of the foot of the perpendicular drawn from the point P(1,-2) on the line y = 2x +1. Also, find the image of P in the line.