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

• $\frac{\mathrm{\pi }}{4}$

• $\frac{\mathrm{\pi }}{3}$

• $\frac{\mathrm{\pi }}{6}$

• $\frac{\mathrm{\pi }}{2}$

If an equilateral triangle is inscribed in the circle x² + y² = a², the lenth of its each side is

• $\sqrt{2}\mathrm{a}$

• $\sqrt{3}\mathrm{a}$

• $\frac{\sqrt{3}}{2}\mathrm{a}$

• $\frac{1}{\sqrt{3}}\mathrm{a}$

If the vertex is (3,0) and the extremities of the latusrectum are (4, 3) and (4, - 3), then the equation of the parabola is

• y² = 4(x - 3)

• x² = 4(y - 3)

• y² = - 4(x + 3)

• x² = - 4(y + 3)

If the line x + 2by + 7 = 0 is a diameter of the circle x² + y² - 6x + zy = 0, then b is equal to

• - 5

• - 3

• 2

• 5

If the line y = 2x+ c is tangent to the parabola y² = 4x, then c is equal to

• $- \frac{1}{2}$

• $\frac{1}{2}$

• $\frac{1}{3}$

• $\frac{1}{4}$

The length of the latus rectum of the ellipse 9x² + 4y² = 1 is

• $\frac{3}{2}$

• $\frac{4}{9}$

• $\frac{8}{3}$

• $\frac{8}{9}$

The number of circles that touch all the straight lines x + y = 4,x - y = - 2 and y = 2 is

• 1

• 2

• 3

• 4

The equation of the normal to the circle x² + y² + 6x + 4y - 3 = 0 at (1, - 2) is

• y + 1 = 0

• y + 2 = 0

• y + 3 = 0

• y - 2 = 0

459.

The limiting points of the co-axial system containing the two circles x² + y² + 2x - 2y + 2 = 0 and 25(x² + y) - 10x - 80y + 65 = 0 are

• (1, - 1), (- 3, - 40)

• $\left(1, - 1\right), \left(- \frac{1}{5}, \frac{8}{5}\right)$

• $\left(- 1, 1\right), \left(\frac{1}{5}, \frac{8}{5}\right)$

• $\left(- \frac{1}{5}, - \frac{8}{5}\right)$

The radical axis of circles x² + y² + 5x + 4y - 5 = 0 and x² + y² - 3x + 5y - 6 = 0 is

• 8y - x + 1 = 0

• 8x - y+ 1 = 0

• 8x - 8y + 1 = 0

• y - 8x + 1 = 0