In numbers from 1 to 100 the digit "0" appears ____________times.
If one of the lines given by the equation 2x2+pxy+3y2=0 coincide with one of those given by 2x2+qxy−3y2=0 and the other lines represented by them are perpendicular, then (a)p=5 (b) p=−5 (c)q=−1 (d) q=1
Prove that the product of the perpendiculars from (α,β) to the pair of lines ax2+2hxy+by2=0 is (a−b)2+4h2aα2+2hαβ+bβ2
Find the equation of the line which satisfy the given conditions : Passing through the point (−4,3)with slope 21.
Find the joint equation of the pair of lines which pass through the origin and are perpendicular to the lines represented the equation y2+3xy−6x+5y−14=0
If p and q are the lengths of perpendiculars from the origin to the lines xcosθ−ysinθ=kcos2θand xsecθ+ycosecθ=k, respectively, prove that p2+4q2=k2.
If p is the length of perpendicular from the origin to the line whose intercepts on the axes are a and b, then show that p21=a21+b21.
Reduce the following equations into intercept form and find their intercepts on the axes.(i) 3x+2y12=0, (ii) 4x3y=6, (iii) 3y+2=0.