If the image of the point (`x_1,y_1`) with respect to the mirror ax+by+c=0 be (`x_2,y_2`), show that `(x_2-x_1)/a=(y_2-y_1)/b=(-2(ax_1+by_1+c))/(a^2+b^2)`.

If Q(h,k) is the image of the point P(x_(1),y_(1)) w.r.to the straight line ax+by+c=0 then prove that (h-x_(1)):a=(k-y_(1)):b=-2(ax_(1)+by_(1)+c):a^(2)+b^(2) or (h-x_(1))/(a)=(k-y_(1))/(b)=(-2(ax_(1)+by_(1)+c))/(a^(2)+b^(2)) and find the image of (1,-2) w.r.t the straight line 2x-3y+5=0

If the points (x_1, y_1), (x_2, y_2) and (x_3, y_3) be collinear, show that: (y_2 - y_3)/(x_2 x_3) + (y_3 - y_1)/(x_3 x_2) + (y_1 - y_2)/(x_1 x_2) = 0

If y=e^(2x)(ax+b), show that y_(2)-4y_(1)+4y=0

Three points A(x_1 , y_1), B (x_2, y_2) and C(x, y) are collinear. Prove that: (x-x_1) (y_2 - y_1) = (x_2 - x_1) (y-y_1) .

If the slope of the curve y=(ax)/(b-x) at the point (1, 1) is 2, then

If the tangents to the parabola y^(2)=4ax at (x_1,y_1) and (x_2,y_2) meet on the axis then x_1y_1+x_2y_2=

If the point x_1+t(x_2-x_1),y_1+t(y_2-y_1) divides the join of (x_1,y_1) and (x_2,y_2) internally then

If y=(ax+b)/(x^(2)+c), prove that (2xy_(1)+y)y_(3)=3(xy_(2)+y_(1))y_(2)

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

The mirror image of the parabola y^(2)=4x in the tangent to the parabola at the point (1,2) is (a)(x-1)^(2)=4(y+1)(b)(x+1)^(2)=4(y+1)(c)(x+1)^(2)=4(y-1) (d) (x-1)^(2)=4(y-1)