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)

74.

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

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

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

• $\frac{36}{5}$

• $\frac{18}{5}$

• $\frac{9}{5}$

• $\frac{1}{5}$

For the variable , the locus of the point of intersection of the lines 3tx - 2y + 6t = 0 and 3x + 2ty - 6 = 0 is

• $\mathrm{the} \mathrm{ellipse} \frac{{\mathrm{x}}^2}{4} + \frac{{\mathrm{y}}^2}{9} = 1$

• $\mathrm{the} \mathrm{ellipse} \frac{{\mathrm{x}}^2}{9} + \frac{{\mathrm{y}}^2}{4} = 1$

• $\mathrm{the} \mathrm{hyperbola} \frac{{\mathrm{x}}^2}{4} - \frac{{\mathrm{y}}^2}{9} = 1$

• $\mathrm{the} \mathrm{hyperbola} \frac{{\mathrm{x}}^2}{9} - \frac{{\mathrm{y}}^2}{4} = 1$

The locus of the mid-points of the chords of an ellipse x² + 4y² = 4 that are drawn from the positive end of the minor axis, is

• a circle with centre $\left(\frac{1}{2}, 0\right)$ and radius 1

• a parabola with focus $\left(\frac{1}{2}, 0\right)$ and directrix x = - 1

• an ellipse with centre $\left(0, \frac{1}{2}\right)$, major axis $\frac{1}{2}$ and minor axis

• a hyperbola with centre $\left(0, \frac{1}{2}\right)$, transverse axis 1 and conjugate axis $\frac{1}{2}$

A point P lies on the circle x² + y² = 169. If Q = (5, 12) and R = (-12, 5) then the $\angle$QPR is

• $\frac{\mathrm{\pi }}{6}$

• $\frac{\mathrm{\pi }}{4}$

• $\frac{\mathrm{\pi }}{3}$

• $\frac{\mathrm{\pi }}{2}$