Ellipse `2x^2+ y^2 = 1` and `y = mx` meet a point A in first quadrant. Normal to the ellipse at P meets x-axis at `(-1/(3sqrt2),0)` and y-axis at `(0, beta)`, then `|beta|` is

Let the line y=mx and the ellipse 2x^(2)+y^(2)=1 intersect at a point P in the first quadrant. If the normal to this ellipse at P meets the co - ordinate axes at (-(1)/(3sqrt2),0) and (0, beta) , then beta is equal to

Let the line y=mx and the ellipse 2x^(2)+y^(2)=1 intersect at a point P in the first quadrant. If the normal to this ellipse at P meets the co - ordinate axes at (-(1)/(3sqrt2),0) and (0, beta) , then beta is equal to

Let the line y=mx and the ellipse 2x^(2)+y^(2)=1 intersect at a point P in the first quadrant. If the normal to this ellipse at P meets the co - ordinate axes at (-(1)/(3sqrt2),0) and (0, beta) , then beta is equal to

The tangent to the ellipse (x^(2))/(25)+(y^(2))/(16)=1 at point P lying in the first quadrant meets x - axis at Q and y - axis at R. If the length QR is minimum, then the equation of this tangent is

Let F_1(x_1,0) and F_2(x_2,0), for x_1 0, be the foci of the ellipse x^2/9+y^2/8=1 Suppose a parabola having vertex at the origin and focus at F_2 intersects the ellipse at point M in the first quadrant and at point N in the fourth quadrant. If the tangents to the ellipse at M and N meet at R and the normal to the parabola at M meets the x-axis at Q, then the ratio of area of the triangle MQR to area of the quadrilateral MF_1 NF_2 is

Let F_1(x_1,0)" and "F_2(x_2,0) , for x_1 lt 0 " and" x_2 gt 0 , be the foci of the ellipse (x^2)/(9)+(y^2)/(8)=1 . Suppose a parabola having vertex at the origin and focus at F_2 intersects the ellipse at point M in the first quadrant and at point N in the fourth quadrant. If the tangents to the ellipse at M and N meet at R and the normal to the parabola at M meets the x-axis at Q, then the ratio of the area of the triangle MQR to area of the quadrilateral MF_1NF_2 is :

Let F_(1)(x_(1),0) and F_(2)(x_(2),0) , for x_(1)lt0 and x_(2)gt0 , be the foci of the ellipse (x^(2))/(9)+(y^(2))/(8)=1 . Suppose a parabola having vertex at the orgin and focus at F_(2) intersects the ellipse at point M in the first quadrant and at point N in the fourth quadrant. If the tangents to the ellipse at M and N meet at R and the normal to the parabola at M meets the x-axis at Q, then the ratio of area of the triangle MQR to area of the quadrilateral MF_(1)NF_(2) is