120784 If the tangent at the point ( \(2 \sec \theta, 3 \tan \theta)\) or the hyperbola \(\frac{x^2}{4}-\frac{y^2}{9}=1\) is parallel to \(3 x-y+\) \(4=0\), then the value of \(\theta\) is

120785 Let the focal chord of the parabola \(P: y^2=4 x\) along the line \(L: y=m x+c, m>0\) meet the parabola at the points \(M\) and \(N\). Let the line \(L\) be a tangent to the hyperbola \(H: x^2-y^2=4\). If \(O\) is the vertex of \(P\) and \(F\) is the focus of \(H\) on the positive \(x\)-axis, then the area of the quadrilateral OMFN is :

120786 Let the eccentricity of the hyperbola \(H: \frac{x^2}{a^2}-\frac{y^2}{b^2}=1\) be \(\sqrt{\frac{5}{2}}\) and length of its latus rectum be \(6 \sqrt{2}\). If \(y=2 x+c\) is a tangent to the hyperbola \(H\), then the value of \(c^2\) is equal to

120787 Let \(P\left(x_0, y_0\right)\) be the point on the hyperbola \(3 x^2\) \(-4 y^2=36\) which is nearest to line \(3 x+2 y=1\). Then \(\sqrt{2}\left(y_0-x_0\right)\) is equal to

120788 The distance between the tangent lines to the hyperbola \(x^2-2 y^2=18\)
which are perpendicular to the line \(y=x\) is

120784 If the tangent at the point ( \(2 \sec \theta, 3 \tan \theta)\) or the hyperbola \(\frac{x^2}{4}-\frac{y^2}{9}=1\) is parallel to \(3 x-y+\) \(4=0\), then the value of \(\theta\) is

120785 Let the focal chord of the parabola \(P: y^2=4 x\) along the line \(L: y=m x+c, m>0\) meet the parabola at the points \(M\) and \(N\). Let the line \(L\) be a tangent to the hyperbola \(H: x^2-y^2=4\). If \(O\) is the vertex of \(P\) and \(F\) is the focus of \(H\) on the positive \(x\)-axis, then the area of the quadrilateral OMFN is :

120786 Let the eccentricity of the hyperbola \(H: \frac{x^2}{a^2}-\frac{y^2}{b^2}=1\) be \(\sqrt{\frac{5}{2}}\) and length of its latus rectum be \(6 \sqrt{2}\). If \(y=2 x+c\) is a tangent to the hyperbola \(H\), then the value of \(c^2\) is equal to

120787 Let \(P\left(x_0, y_0\right)\) be the point on the hyperbola \(3 x^2\) \(-4 y^2=36\) which is nearest to line \(3 x+2 y=1\). Then \(\sqrt{2}\left(y_0-x_0\right)\) is equal to

120788 The distance between the tangent lines to the hyperbola \(x^2-2 y^2=18\)
which are perpendicular to the line \(y=x\) is

120784 If the tangent at the point ( \(2 \sec \theta, 3 \tan \theta)\) or the hyperbola \(\frac{x^2}{4}-\frac{y^2}{9}=1\) is parallel to \(3 x-y+\) \(4=0\), then the value of \(\theta\) is

120785 Let the focal chord of the parabola \(P: y^2=4 x\) along the line \(L: y=m x+c, m>0\) meet the parabola at the points \(M\) and \(N\). Let the line \(L\) be a tangent to the hyperbola \(H: x^2-y^2=4\). If \(O\) is the vertex of \(P\) and \(F\) is the focus of \(H\) on the positive \(x\)-axis, then the area of the quadrilateral OMFN is :

120786 Let the eccentricity of the hyperbola \(H: \frac{x^2}{a^2}-\frac{y^2}{b^2}=1\) be \(\sqrt{\frac{5}{2}}\) and length of its latus rectum be \(6 \sqrt{2}\). If \(y=2 x+c\) is a tangent to the hyperbola \(H\), then the value of \(c^2\) is equal to

120787 Let \(P\left(x_0, y_0\right)\) be the point on the hyperbola \(3 x^2\) \(-4 y^2=36\) which is nearest to line \(3 x+2 y=1\). Then \(\sqrt{2}\left(y_0-x_0\right)\) is equal to

120788 The distance between the tangent lines to the hyperbola \(x^2-2 y^2=18\)
which are perpendicular to the line \(y=x\) is

120784 If the tangent at the point ( \(2 \sec \theta, 3 \tan \theta)\) or the hyperbola \(\frac{x^2}{4}-\frac{y^2}{9}=1\) is parallel to \(3 x-y+\) \(4=0\), then the value of \(\theta\) is

120785 Let the focal chord of the parabola \(P: y^2=4 x\) along the line \(L: y=m x+c, m>0\) meet the parabola at the points \(M\) and \(N\). Let the line \(L\) be a tangent to the hyperbola \(H: x^2-y^2=4\). If \(O\) is the vertex of \(P\) and \(F\) is the focus of \(H\) on the positive \(x\)-axis, then the area of the quadrilateral OMFN is :

120786 Let the eccentricity of the hyperbola \(H: \frac{x^2}{a^2}-\frac{y^2}{b^2}=1\) be \(\sqrt{\frac{5}{2}}\) and length of its latus rectum be \(6 \sqrt{2}\). If \(y=2 x+c\) is a tangent to the hyperbola \(H\), then the value of \(c^2\) is equal to

120787 Let \(P\left(x_0, y_0\right)\) be the point on the hyperbola \(3 x^2\) \(-4 y^2=36\) which is nearest to line \(3 x+2 y=1\). Then \(\sqrt{2}\left(y_0-x_0\right)\) is equal to

120788 The distance between the tangent lines to the hyperbola \(x^2-2 y^2=18\)
which are perpendicular to the line \(y=x\) is

120784 If the tangent at the point ( \(2 \sec \theta, 3 \tan \theta)\) or the hyperbola \(\frac{x^2}{4}-\frac{y^2}{9}=1\) is parallel to \(3 x-y+\) \(4=0\), then the value of \(\theta\) is

120785 Let the focal chord of the parabola \(P: y^2=4 x\) along the line \(L: y=m x+c, m>0\) meet the parabola at the points \(M\) and \(N\). Let the line \(L\) be a tangent to the hyperbola \(H: x^2-y^2=4\). If \(O\) is the vertex of \(P\) and \(F\) is the focus of \(H\) on the positive \(x\)-axis, then the area of the quadrilateral OMFN is :

120786 Let the eccentricity of the hyperbola \(H: \frac{x^2}{a^2}-\frac{y^2}{b^2}=1\) be \(\sqrt{\frac{5}{2}}\) and length of its latus rectum be \(6 \sqrt{2}\). If \(y=2 x+c\) is a tangent to the hyperbola \(H\), then the value of \(c^2\) is equal to

120787 Let \(P\left(x_0, y_0\right)\) be the point on the hyperbola \(3 x^2\) \(-4 y^2=36\) which is nearest to line \(3 x+2 y=1\). Then \(\sqrt{2}\left(y_0-x_0\right)\) is equal to

120788 The distance between the tangent lines to the hyperbola \(x^2-2 y^2=18\)
which are perpendicular to the line \(y=x\) is