The lines y = 2x + $\sqrt{76}$ and 2y + x = 8 touch the ellipse x² + y² = 1. If the point of $\frac{{\mathrm{x}}^2}{16} + \frac{{\mathrm{y}}^2}{12} = 1$ intersection of these two lines lie on a circle, whose centre coincides with the centre of that ellipse, then the equation of that circle is

• ${\mathrm{x}}^2 + {\mathrm{y}}^2 = 28$

• ${\mathrm{x}}^2 + {\mathrm{y}}^2 = 12$

• ${\mathrm{x}}^2 + {\mathrm{y}}^2 = 12$

• ${\mathrm{x}}^2 + {\mathrm{y}}^2 = {\left(4 + \sqrt{8}\right)}^2$

If lx + my = 1 is a normal to the hyperbola $\frac{{\mathrm{x}}^2}{{\mathrm{a}}^2} - \frac{{\mathrm{y}}^2}{{\mathrm{b}}^2} = 1$, then a²m² - b²l² = ?

• $\frac{{\mathrm{m}}^2}{{\mathrm{l}}^2}{\left({\mathrm{a}}^2 + {\mathrm{b}}^2\right)}^2$

• l² + m²(a² + b²)²

• $\frac{{\mathrm{l}}^2}{{\mathrm{m}}^2}{\left({\mathrm{a}}^2 + {\mathrm{b}}^2\right)}^2$

• l²m²(a² + b²)²

$\frac{\mathrm{x} - 1}{3\mathrm{x} + 4} < \frac{\mathrm{x} - 3}{3\mathrm{x} - 2} \mathrm{holds}, \mathrm{for} \mathrm{all} \mathrm{x} \mathrm{in} \mathrm{the} \mathrm{internal}$

• $\left(\frac{ - 4}{3}, \frac{2}{3}\right)$

• $\left(\infty , \frac{ - 5}{4}\right)$

• $\left(\frac{3}{3}, \infty \right)$

• $\left( - \infty , \frac{ - 5}{4}\right) \cup \left(\frac{3}{4}, - \infty \right)$

If the point of intersection of the tangents drawn at the points where the line 5x + y + 1 = 0 cuts the circle x² + y² - zx - 6y - 8 = 0 is (a, b), then 5a + b =

• 3

•  - 44

•  - 1

• 4

If 2kx + 3y - 1 = 0, 2x + y + 5 = 0 are conjugate lines with respect to the circle x² + y² - 2x - 4y - 4 = 0, then k =

• 3

• 4

• 1

• 2

The equations of the parabola whose axis is parallel to the X-axis and which passes through the points (- 2, 1), (1, 2)(- 1, 3) is

• $18{\mathrm{y}}^2 - 12\mathrm{x} - 21\mathrm{y} - 21 = 0$

• $5{\mathrm{y}}^2 + 2\mathrm{x} - 21\mathrm{y} + 20 = 0$

• $15{\mathrm{y}}^2 + 12\mathrm{x} - 11\mathrm{y} + 20 = 0$

• $25{\mathrm{y}}^2 - 2\mathrm{x} - 65\mathrm{y} + 36 = 0$

The angle between the two circles, each passing through the centre of the other is

• $\frac{2\mathrm{\pi }}{3}$

• $\frac{\mathrm{\pi }}{2}$

• $\frac{\mathrm{\pi }}{6}$

• $\mathrm{\pi }$

$\mathrm{If} {\log }_{\frac{1}{\sqrt{3}}}\left\{\frac{{\left|\mathrm{z}\right|}^2 - \left|\mathrm{z}\right| + 1}{2 + \left|\mathrm{z}\right|}\right\} > - 2, \mathrm{then} \mathrm{z} \mathrm{lies} \mathrm{inside}$

• a triangle

• an ellipse

• a circle

• a square

A circle having centre at the origin passes through the three vertices of an equilateral triangle the length of its median being 9 units. Then the equation of that circle is

• ${\mathrm{x}}^2 + {\mathrm{y}}^2 = 9$

• ${\mathrm{x}}^2 + {\mathrm{y}}^2 = 18$

• ${\mathrm{x}}^2 + {\mathrm{y}}^2 = 36$

• ${\mathrm{x}}^2 + {\mathrm{y}}^2 = 81$

A line parallel to the straight line 2x – y = 0 is tangent to the hyperbola $\frac{{\mathrm{x}}^2}{4} - \frac{{\mathrm{y}}^2}{2} = 1$  at the point (x₁, y₁). Then ${{\mathrm{x}}_1}^2 + 5{{\mathrm{y}}_1}^2$ is equal to :

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• 5

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