120814 A line parallel to the straight line \(2 x-y=0\) is tangent to be hyperbola \(\frac{x^2}{4}-\frac{y^2}{2}=1\) at the point \(\left(x_1, y_1\right)\). Then \(x_1{ }^2+5 y_1{ }^2\) is equal to

120815 Consider a hyperbola \(\mathrm{H}: \mathrm{x}^2-2 \mathrm{y}^2=4\). Let the tangent at a point \(P(4, \sqrt{6})\) meet the \(x\)-axis at \(Q\) and latus rectum at \(R\left(x_1, y_1\right), x_1>0\). If \(F\) is a focus of \(H\) which is nearer to the point \(P\), then the area of \(\triangle \mathrm{QFR}\) is equal to

120816 The locus of the mid-points of the chord of the circle, \(x^2+y^2=25\) which is tangent to the hyperbola, \(\frac{x^2}{9}-\frac{y^2}{16}=1\) is

120817 Let line \(L: 2 x+y=k, k>0\) be a tangent to the hyperbola \(x^2-y^2=3\). If \(L\) is also a tangent to the parabola \(y^2=\alpha x\), then \(\alpha\) is equal to

120818 The point \(P(-2 \sqrt{6}, \sqrt{3})\) lies on the hyperbola \(\frac{x^2}{a^2}-\frac{y^2}{b^2}=1\) having eccentricity \(\frac{\sqrt{5}}{2}\). If the tangent and normal at \(P\) to the hyperbola intersect its conjugate axis at the point \(Q\) and \(R\) respectively, then \(Q R\) is equal to

120814 A line parallel to the straight line \(2 x-y=0\) is tangent to be hyperbola \(\frac{x^2}{4}-\frac{y^2}{2}=1\) at the point \(\left(x_1, y_1\right)\). Then \(x_1{ }^2+5 y_1{ }^2\) is equal to

120815 Consider a hyperbola \(\mathrm{H}: \mathrm{x}^2-2 \mathrm{y}^2=4\). Let the tangent at a point \(P(4, \sqrt{6})\) meet the \(x\)-axis at \(Q\) and latus rectum at \(R\left(x_1, y_1\right), x_1>0\). If \(F\) is a focus of \(H\) which is nearer to the point \(P\), then the area of \(\triangle \mathrm{QFR}\) is equal to

120816 The locus of the mid-points of the chord of the circle, \(x^2+y^2=25\) which is tangent to the hyperbola, \(\frac{x^2}{9}-\frac{y^2}{16}=1\) is

120817 Let line \(L: 2 x+y=k, k>0\) be a tangent to the hyperbola \(x^2-y^2=3\). If \(L\) is also a tangent to the parabola \(y^2=\alpha x\), then \(\alpha\) is equal to

120818 The point \(P(-2 \sqrt{6}, \sqrt{3})\) lies on the hyperbola \(\frac{x^2}{a^2}-\frac{y^2}{b^2}=1\) having eccentricity \(\frac{\sqrt{5}}{2}\). If the tangent and normal at \(P\) to the hyperbola intersect its conjugate axis at the point \(Q\) and \(R\) respectively, then \(Q R\) is equal to

120814 A line parallel to the straight line \(2 x-y=0\) is tangent to be hyperbola \(\frac{x^2}{4}-\frac{y^2}{2}=1\) at the point \(\left(x_1, y_1\right)\). Then \(x_1{ }^2+5 y_1{ }^2\) is equal to

120815 Consider a hyperbola \(\mathrm{H}: \mathrm{x}^2-2 \mathrm{y}^2=4\). Let the tangent at a point \(P(4, \sqrt{6})\) meet the \(x\)-axis at \(Q\) and latus rectum at \(R\left(x_1, y_1\right), x_1>0\). If \(F\) is a focus of \(H\) which is nearer to the point \(P\), then the area of \(\triangle \mathrm{QFR}\) is equal to

120816 The locus of the mid-points of the chord of the circle, \(x^2+y^2=25\) which is tangent to the hyperbola, \(\frac{x^2}{9}-\frac{y^2}{16}=1\) is

120817 Let line \(L: 2 x+y=k, k>0\) be a tangent to the hyperbola \(x^2-y^2=3\). If \(L\) is also a tangent to the parabola \(y^2=\alpha x\), then \(\alpha\) is equal to

120818 The point \(P(-2 \sqrt{6}, \sqrt{3})\) lies on the hyperbola \(\frac{x^2}{a^2}-\frac{y^2}{b^2}=1\) having eccentricity \(\frac{\sqrt{5}}{2}\). If the tangent and normal at \(P\) to the hyperbola intersect its conjugate axis at the point \(Q\) and \(R\) respectively, then \(Q R\) is equal to

120814 A line parallel to the straight line \(2 x-y=0\) is tangent to be hyperbola \(\frac{x^2}{4}-\frac{y^2}{2}=1\) at the point \(\left(x_1, y_1\right)\). Then \(x_1{ }^2+5 y_1{ }^2\) is equal to

120815 Consider a hyperbola \(\mathrm{H}: \mathrm{x}^2-2 \mathrm{y}^2=4\). Let the tangent at a point \(P(4, \sqrt{6})\) meet the \(x\)-axis at \(Q\) and latus rectum at \(R\left(x_1, y_1\right), x_1>0\). If \(F\) is a focus of \(H\) which is nearer to the point \(P\), then the area of \(\triangle \mathrm{QFR}\) is equal to

120816 The locus of the mid-points of the chord of the circle, \(x^2+y^2=25\) which is tangent to the hyperbola, \(\frac{x^2}{9}-\frac{y^2}{16}=1\) is

120817 Let line \(L: 2 x+y=k, k>0\) be a tangent to the hyperbola \(x^2-y^2=3\). If \(L\) is also a tangent to the parabola \(y^2=\alpha x\), then \(\alpha\) is equal to

120818 The point \(P(-2 \sqrt{6}, \sqrt{3})\) lies on the hyperbola \(\frac{x^2}{a^2}-\frac{y^2}{b^2}=1\) having eccentricity \(\frac{\sqrt{5}}{2}\). If the tangent and normal at \(P\) to the hyperbola intersect its conjugate axis at the point \(Q\) and \(R\) respectively, then \(Q R\) is equal to

120814 A line parallel to the straight line \(2 x-y=0\) is tangent to be hyperbola \(\frac{x^2}{4}-\frac{y^2}{2}=1\) at the point \(\left(x_1, y_1\right)\). Then \(x_1{ }^2+5 y_1{ }^2\) is equal to

120815 Consider a hyperbola \(\mathrm{H}: \mathrm{x}^2-2 \mathrm{y}^2=4\). Let the tangent at a point \(P(4, \sqrt{6})\) meet the \(x\)-axis at \(Q\) and latus rectum at \(R\left(x_1, y_1\right), x_1>0\). If \(F\) is a focus of \(H\) which is nearer to the point \(P\), then the area of \(\triangle \mathrm{QFR}\) is equal to

120816 The locus of the mid-points of the chord of the circle, \(x^2+y^2=25\) which is tangent to the hyperbola, \(\frac{x^2}{9}-\frac{y^2}{16}=1\) is

120817 Let line \(L: 2 x+y=k, k>0\) be a tangent to the hyperbola \(x^2-y^2=3\). If \(L\) is also a tangent to the parabola \(y^2=\alpha x\), then \(\alpha\) is equal to

120818 The point \(P(-2 \sqrt{6}, \sqrt{3})\) lies on the hyperbola \(\frac{x^2}{a^2}-\frac{y^2}{b^2}=1\) having eccentricity \(\frac{\sqrt{5}}{2}\). If the tangent and normal at \(P\) to the hyperbola intersect its conjugate axis at the point \(Q\) and \(R\) respectively, then \(Q R\) is equal to