Let l be a line which is normal to the curve `y = 2x^2 + x + 2` at a point P on the curve. If the point Q( 6. 4) lies on the line l and O is origin, then the area of thetriangle· OPQ is equal to ____ _

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

Find the equation of normal of the curve 2y= 7x - 5x^(2) at those points at which the curve intersects the line x = y.

Let P is a point on the line y+2x=2 and Q and R are two points on the line 3y+6x=3 . If the triangle PQR is an equilateral triangle, then its area (in sq. units) is equal to

The y-intercept of the line normal to the curve y^2=x^2+33(y gt 0) at the point with abscissa 4 is :

If the curve y=px^(2)+qx+r passes through the point (1,2) and the line y=x touches it at the origin,then the values of p,q and r are

If the curve y=px^(2)+qx+r passes through the point (1,2) and the line y=x touches it at the origin,then the values of p,q and r are

Let the line L : sqrt(2)x y =alpha pass through the point of the intersection P (in the first quadrant) of the circle x^2 y^2 = 3 and the parabola x^2 = 2y . Let the line L touch two circles C_1 and C_2 of equal radius 2 sqrt 3 . If the centres Q_1 and Q_2 of the circles C_1 and C_2 lie on the y-axis, then the square of the area of the triangle PQ_1Q_2 is equal to ___________.

The slope of the tangent to the curve y=x^(2) -x at the point where the line y = 2 cuts the curve in the first quadrant, is