If `x^2+2y^2=1` and `x+y=1` intersect and line joining them subtends an angle at the origin , then find angle

If the portion of the line ax - by - 1 = 0 , intercepted between the lines ax + y + 1 = 0 and x + by = 0 subtends a right angle at the origin , then find value of 4a +b^2 + (b+1)^2 .

A straight line is drawn from the point (1,0) to the curve x^(2)+y^(2)+6x-10y+1=0, such that the intercept made on it by the curve subtends a right angle at the origin.Find the equations of the line.

" Eccentricity of ellipse " (x^(2))/(a^(2))+(y^(2))/(b^(2))=1 " such that the line joining the foci subtends a right angle only at two points on ellipse is "

If the line x+2y+4=0 cutting the ellipse (x^(2))/(a^(2))+(y^(2))/(b^(2))=1 in points whose eccentric angies are 30^(@) and 60^(@) subtends right angle at the origin then its equation is

If the curve x^2 + 2y^2 = 2 intersects the line x + y = 1 at two points P and Q, then the angle subtended by the line segment PQ at the origin is :

Line x+ ay =1 cuts the circle x^(2)+y^(2)-4x-2y+1=0 at two points A and B such that Chord AB subtends an angle of 90^(@) at origin then sum of possible values of a is

Line x+ ay =1 cuts the circle x^(2)+y^(2)-4x-2y+1=0 at two points A and B such that Chord AB subtends an angle of 90^(@) at origin then sum of possible values of a is

If the common chord of the circles x^(2)+(y-2)^(2)=16 and x^(2)+y^(2)=16 subtend a angle at the origin then lambda is equal to

A line intersects the ellipe (x^(2))/(a^(2))+(y^(2))/(b^(2))=1 at P and Q and the parabola y^(2)=4d(x+a) at R and S. The line segment PQ subtends a right angle at the centre of the ellipse. Find the locus of the point intersection of the tangents to the parabola at R and S.

If the line segment joining the points P(x_1, y_1)a n d Q(x_2, y_2) subtends an angle alpha at the origin O, prove that : O PdotO Q cosalpha=x_1x_2+y_1y_2dot