The tangent to the curve `y=e^x` drawn at the point `(c,e^c)` intersects the line joining the points `(c-1,e^(c-1))` and `(c+1,e^(c+1))`

The tangent to the curve y=xe^(x^2) passing through the point (1,e) also passes through the point

The equation of the normal to the curve y=e^(-2|x| at the point where the curve cuts the line x=-1/2 , is

The tangent drawn at the point (0, 1) on the curve y=e^(2x) meets the x-axis at the point -

The equation of the tangent to the curve y=b e^(-x//a) at the point where it crosses the y-axis is (a)x/a-y/b=1 (b) a x+b y=1 (c)a x-b y=1 (d) x/a+y/b=1

At what point is the tangent to the curve f(x)=log x parallel to the chord joining the points A(1, 0) and B(e, 1) ?

If the z-coordinate of a point C on the line joining the points A(2, 2, 1) and B(5, 1, -2) is (-1), then x-coordinate of C is -

Determine the constant c such that the straight line joining the points (0,3) and (5-2) is a tangent to the curve y=(c )/(x+1) .

A curve y=f(x) passes through point P(1,1) . The normal to the curve at P is a (y-1)+(x-1)=0 . If the slope of the tangent at any point on the curve is proportional to the ordinate of the point, then the equation of the curve is (a) ( b ) (c) y=( d ) e^(( e ) (f) K(( g ) (h) x-1( i ))( j ))( k ) (l) (m) (b) ( n ) (o) y=( p ) e^(( q ) (r) K e (s))( t ) (u) (v) (c) ( d ) (e) y=( f ) e^(( g ) (h) K(( i ) (j) x-2( k ))( l ))( m ) (n) (o) (d) None of these

Show that the equation of the straight line throught (alpha,beta) and through the point of intersection of the lines a_(1)x+b_(1)y+c_(1)=0 anda_(2)x+b_(2)y+c_(2)=0 is (a_(1)x+b_(1)y+c_(1))/(a_(1)alpha+b_(1)beta+c_(1))=(a_(2)x+b_(2)y+c_(2))/(a_(2)alpha+b_(2)beta+c_(2))

Find a pair of curves such that (a) the tangents drawn at points with equal abscissas intersect on the y-axis. (b) the normal drawn at points with equal abscissas intersect on the x-axis. (c) one curve passes through (1,1) and other passes through (2, 3).