If y = 4x + 3 is parallel to a tangent to the parabola y² = 12x, then its distance from the normal parallel to the given line is

• $\frac{213}{\sqrt{17}}$

• $\frac{219}{\sqrt{17}}$

• $\frac{211}{\sqrt{17}}$

• $\frac{210}{\sqrt{17}}$

Let the equation of an ellipse be $\frac{{\mathrm{x}}^2}{144} + \frac{{\mathrm{y}}^2}{25} = 1$. Then, the radius of the circle with centre (0, $\sqrt{2}$) and passing through the foci of the ellipse is

• 9

• 7

• 11

• 5

The value of $\mathrm{\lambda }$ for which the curve (7x + 5)² + (7y + 3)² = ${\mathrm{\lambda }}^2$(4x + 3y - 24)² represents a parabola is

• $\pm \frac{6}{5}$

• $\pm \frac{7}{5}$

• $\pm \frac{1}{5}$

• $\pm \frac{2}{5}$

The equation of the common tangent with positive slope to the parabola y² = 8$\sqrt{3}$x and the hyperbola 4x² - y² = 4 is

• $\mathrm{y} = \sqrt{6}\mathrm{x} + \sqrt{2}$

• $\mathrm{y} = \sqrt{6}\mathrm{x} - \sqrt{2}$

• $\mathrm{y} = \sqrt{3}\mathrm{x} + \sqrt{2}$

• $\mathrm{y} = \sqrt{3}\mathrm{x} - \sqrt{2}$

The point on the parabola y² = 64x which is nearest to the line 4x + 3y + 35 = 0 has coordinates

• (9, - 24)

• (1, 81)

• (4, - 16)

• (- 9, - 24)

Let z₁, z₂ be two fixed complex numbers in the argand plane and z be an arbitrary point satisfying $\left|\mathrm{z} - {\mathrm{z}}_1\right| + \left|\mathrm{z} - {\mathrm{z}}_2\right| = 2\left|{\mathrm{z}}_1 - {\mathrm{z}}_2\right|$ Then, the locus of z will be

• an ellipse

• a straight line joining z₁ and z₂

• a parabola

• a bisector of the line segment joining z₁ and z₂

77.

Let z, be a fixed point on the circle of radius 1 centered at the origin in the Argand plane and ${\mathrm{z}}_1 \ne \pm 1$. Consider an equilateral triangle inscribed in the circle with z₁, z₂, z₃ as the vertices taken in the counterclockwise direction. Then, z₁z₂z₃ is equal to

• z₁²

• z₁³

• z₁⁴

• z₁

The equation of hyperbola whose coordinates of the foci are (± 8, 0) and the length of latusrectum is 24 units, is

• 3x² - y² = 48

• 4x² - y² = 48

• x² - 3y² = 48

• x² - 4y² = 48

If the circle x² + y² + 2gx + 2fy + c = 0 cuts the three circles x² + y² - 5 = 0, x² + y² - 8x - 6y + 10 = 0 and x² + y² - 4x + 2y - 2 = 0 at the extremities of  their diameters, then

• c = - 5

• fg = 147/25

• g + 2f = c + 2

• 4f = 3g

Lines x + y = 1 and 3y = x + 3 intersect the ellipse x² + 9y² = 9 at the points P,Q and R. The area of the $∆$PQR is

• $\frac{36}{5}$

• $\frac{18}{5}$

• $\frac{9}{5}$

• $\frac{1}{5}$