120755 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 \(\mathrm{H}\), then the value of \(\mathrm{c}^2\) is equal to

120756 If the eccentricity of hyperbola \(x^2-y^2 \sec ^2 \alpha=\) 5 is \(\sqrt{3}\) times the eccentricity of the ellipse \(x^2\) \(\sec ^2 \alpha+y^2=25\), then value of \(\alpha\) is

120757 On a rectangular hyperbola \(\mathrm{x}^2-\mathrm{y}^2=\mathrm{a}^2, \mathrm{a}>0\), three points \(A, B, C\) are taken as follows: \(A=\) \((-a, 0) ; B\) and \(C\) are placed symmetrically with respect to the \(x\)-axis on the branch of the hyperbola not containing A. Suppose that the triangle \(A B C\) is equilateral. If the side-length of the triangle \(A B C\) is \(k\), then \(k\) lies in the interval

120758 Statement 1 : The eccentricity of the hyperbola \(9 x^2-16 y^2-72 x+96 y-144=0\) is \(\frac{5}{4}\)
Statement 2: The eccentricity of the hyperbola \(\frac{x^2}{a^2}-\frac{y^2}{b^2}=1\) is \(\sqrt{1+\frac{b^2}{a^2}}\)

120755 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 \(\mathrm{H}\), then the value of \(\mathrm{c}^2\) is equal to

120756 If the eccentricity of hyperbola \(x^2-y^2 \sec ^2 \alpha=\) 5 is \(\sqrt{3}\) times the eccentricity of the ellipse \(x^2\) \(\sec ^2 \alpha+y^2=25\), then value of \(\alpha\) is

120757 On a rectangular hyperbola \(\mathrm{x}^2-\mathrm{y}^2=\mathrm{a}^2, \mathrm{a}>0\), three points \(A, B, C\) are taken as follows: \(A=\) \((-a, 0) ; B\) and \(C\) are placed symmetrically with respect to the \(x\)-axis on the branch of the hyperbola not containing A. Suppose that the triangle \(A B C\) is equilateral. If the side-length of the triangle \(A B C\) is \(k\), then \(k\) lies in the interval

120758 Statement 1 : The eccentricity of the hyperbola \(9 x^2-16 y^2-72 x+96 y-144=0\) is \(\frac{5}{4}\)
Statement 2: The eccentricity of the hyperbola \(\frac{x^2}{a^2}-\frac{y^2}{b^2}=1\) is \(\sqrt{1+\frac{b^2}{a^2}}\)

120755 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 \(\mathrm{H}\), then the value of \(\mathrm{c}^2\) is equal to

120756 If the eccentricity of hyperbola \(x^2-y^2 \sec ^2 \alpha=\) 5 is \(\sqrt{3}\) times the eccentricity of the ellipse \(x^2\) \(\sec ^2 \alpha+y^2=25\), then value of \(\alpha\) is

120757 On a rectangular hyperbola \(\mathrm{x}^2-\mathrm{y}^2=\mathrm{a}^2, \mathrm{a}>0\), three points \(A, B, C\) are taken as follows: \(A=\) \((-a, 0) ; B\) and \(C\) are placed symmetrically with respect to the \(x\)-axis on the branch of the hyperbola not containing A. Suppose that the triangle \(A B C\) is equilateral. If the side-length of the triangle \(A B C\) is \(k\), then \(k\) lies in the interval

120758 Statement 1 : The eccentricity of the hyperbola \(9 x^2-16 y^2-72 x+96 y-144=0\) is \(\frac{5}{4}\)
Statement 2: The eccentricity of the hyperbola \(\frac{x^2}{a^2}-\frac{y^2}{b^2}=1\) is \(\sqrt{1+\frac{b^2}{a^2}}\)

120755 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 \(\mathrm{H}\), then the value of \(\mathrm{c}^2\) is equal to

120756 If the eccentricity of hyperbola \(x^2-y^2 \sec ^2 \alpha=\) 5 is \(\sqrt{3}\) times the eccentricity of the ellipse \(x^2\) \(\sec ^2 \alpha+y^2=25\), then value of \(\alpha\) is

120757 On a rectangular hyperbola \(\mathrm{x}^2-\mathrm{y}^2=\mathrm{a}^2, \mathrm{a}>0\), three points \(A, B, C\) are taken as follows: \(A=\) \((-a, 0) ; B\) and \(C\) are placed symmetrically with respect to the \(x\)-axis on the branch of the hyperbola not containing A. Suppose that the triangle \(A B C\) is equilateral. If the side-length of the triangle \(A B C\) is \(k\), then \(k\) lies in the interval

120758 Statement 1 : The eccentricity of the hyperbola \(9 x^2-16 y^2-72 x+96 y-144=0\) is \(\frac{5}{4}\)
Statement 2: The eccentricity of the hyperbola \(\frac{x^2}{a^2}-\frac{y^2}{b^2}=1\) is \(\sqrt{1+\frac{b^2}{a^2}}\)