If the equations `y=m x+c` and `xcosalpha+ysinalpha=p` represent the same straight line, then (a)`p=csqrt(1+m^2)` (b) `c=psqrt(1+m^2)` (c)`c p=sqrt(1+m^2)` (d) `p^2+c^2+m^2=1`

If the straight line xcosalpha+ysinalpha=p touches the curve x y=a^2, then prove that p^2=4a^2cosalphasinalphadot

If z=x+i y , then the equation |(2z-i)/(z+1)|=m does not represents a circle, when m is (a) 1/2 (b). 1 (c). 2 (d). 3

If the equation (2p-1)^(2)-p(p+1)=0 is expressed in the form ap^(2)+bp+c=0(ane0) , then b=-5.

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 (a) 1/(sqrt(3)) (b) -1/(sqrt(3)) (c) 1/(sqrt(2)) (d) none of these

If y=m_1x+c and y=m_2x+c are two tangents to the parabola y^2+4a(x+c)=0 , then

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 (a) 2sqrt(3)-3 (b) 2sqrt(2)-1 (c) 2sqrt(2)-1 (d) 1

If the straight line xcosalpha+ysinalpha=p touches the curve (x^2)/(a^2)+(y^2)/(b^2)=1 , then prove that a^2cos^2alpha-b^2sin^2alpha=p^2 .

If a , b and c are in A P , then the straight line a x+b y+c=0 will always pass through a fixed point whose coordinates are (a) (1,2) (b) (1,-2) (c) (2,3) (d) (0,0)

If x cos alpha+ y sin alpha=p , where p=(-sin^(2) alpha)/(cos alpha) be a straight, line, prove that the perpendicular p_(1), p_(2) and p_(3) on this line drawn from the point (m^(2), 2m), (mm', m+m' and) ((m')^(2), 2m') respectively, are in geometric progression. (m gt 0, m gt 0, 0 lt alpha lt 90^(@)) .

If the ratio of A.M. of two numbers x and y to their G.M. is p : q, show that, x : y = (p + sqrt(p^(2) - q^(2))) : (p- sqrt(p^(2) - q^(2)))