Prove that the lengths of the perpendicular from the points `(m^(2), 2m), (mm', m+m'), " and " (m'^(2), 2m')` to the line x+y+1=0 are in GP.

Prove that the lengths of the perpendiculars from the points (m^2,2m),(m m^(prime),m+m^(prime)), and (m^('2),2m^(prime)) to the line x+y+1=0 are in GP.

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^(@)) .

The perpendicular from the origin to the line y = mx + c meets it at the point (-1, 2) . Find the values of m and c.

The coordinates of the mid-point of the line segment joining the points (l, 2m) and (-l +2m, 2l -2m) are

If the straight line joining the points (4, -3, 2) and (3, -1, 5) is perpendicular to the straight line joining the points (m, -2, 1) and (7, 3, -2), then the value of m is -

Normal drawn to y^2=4a x at the points where it is intersected by the line y=m x+c intersect at P . The foot of the another normal drawn to the parabola from the point P is (a) (a/(m^2),-(2a)/m) (b) ((9a)/m ,-(6a)/m) (c) (a m^2,-2a m) (d) ((4a)/(m^2),-(4a)/m)

If the tangent to the curve y^(2) = x^(3) at the point (m^(2), m^(3)) is also the normal to the curve at (M^(2), M^(3)) , then the value of mM is-

Show that the area of the triangle formed by the lines y = m_(1) x + c_(1) , y = m_(2) x + c_(2) " and " x = 0 " is " ((c_(1) - c_(2))^(2))/(2|m_(1) - m_(2)|)

The points P divides the line - segment joining the points A(x_(1),y_(1))andB(x_(2),y_(2)) in the ratio m : n . If P lies on the line ax+by +c=0 , prove that, (m)/(n)=-(ax_(1)+by_(1)+c)/(ax_(2)+by_(2)+c) .