If the point `P(a ,a^2)` lies completely inside the triangle formed by the lines `x=0,y=0,` and `x+y=2,` then find the exhaustive range of values of `a` is (A) `(0,1)` (B) `(1,sqrt2)` (C) `(sqrt2 -1,1)` (D) `(sqrt2 -1,2)`

Find the coordinates of the centroid of the triangle, formed by the lines x+2y-5=0, y+2x-7=0 and x-y+1=0 .

If the circle x^2+y^2+2a_1x+c=0 lies completely inside the circle x^2+y^2+2a_2x+c=0 then

Find the obtuse angle between the lines x-y+2=0 and (2-sqrt(3))x-y+1=0 .

The area of the triangle formed by the lines y=a x ,x+y-a=0 and the y-axis is (a) 1/(2|1+a|) (b) 1/(|1+a|) (c) 1/2|a/(1+a)| (d) (a^2)/(2|1+a|)

Find alpha if (alpha, alpha^2) lies inside the triangle having sides along the lines 2x+3y=1, x+2y-3=0, 6y=5x-1.

If the lines x+(a-1)y+1=0 and 2x+a^2y-1=0 are perpendicular, then find the value of adot

Statement 1: The value of alpha for which the point (alpha,alpha^2) lies inside the triangle formed by the lines x=0,x+y=2 and 3y=x is (0,1)dot Statement 2: The parabola y=x^2 meets the line x+y=2 at (1,1)dot (a) Both the statements are true, and Statement-1 is the correct explanation of Statement 2. (b)Both the statements are true, and Statement-1 is not the correct explanation of Statement 2. (c) Statement 1 is true and Statement 2 is false. (d) Statement 1 is false and Statement 2 is true.

If the angle theta between the line (x+1)/1=(y-1)/2=(z-2)/2 and the plane 2x-y+sqrt(pz)+4=0 is such that sintheta=1/3 , then the values of p is (A) 0 (B) 1/3 (C) 2/3 (D) 5/3

The ratio of the area of triangle inscribed in ellipse x^2/a^2+y^2/b^2=1 to that of triangle formed by the corresponding points on the auxiliary circle is 0.5. Then, find the eccentricity of the ellipse. (A) 1/2 (B) sqrt3/2 (C) 1/sqrt2 (D) 1/sqrt3

If x = 3 + 2 sqrt2 , then find the value of x + (1)/(x)