$$\begin{gathered} \mathrm{Let} \mathrm{x} = 4 \mathrm{be} \mathrm{a} \mathrm{directrix} \mathrm{to} \mathrm{an} \mathrm{ellipse} \mathrm{whose} \mathrm{centre} \mathrm{is} \mathrm{at} \mathrm{the} \mathrm{origin} \mathrm{and} \mathrm{its} \mathrm{eccentricity} \mathrm{is} 21. \\ \mathrm{If} \mathrm{P}(1, \mathrm{\beta }), \mathrm{\beta } > 0 \mathrm{is} \mathrm{a} \mathrm{point} \mathrm{on} \mathrm{this} \mathrm{ellipse}, \mathrm{then} \mathrm{the} \mathrm{equation} \mathrm{of} \mathrm{the} \mathrm{normal} \mathrm{to} \mathrm{it} \mathrm{at} \mathrm{P} \mathrm{is} \end{gathered}$$

• 7x - 4y = 1

• 4x - 2y = 1

• 8x - 2y = 5

• 4x - 3y = 2

$$\begin{gathered} \mathrm{Let} \mathrm{PQ} \mathrm{be} \mathrm{a} \mathrm{diameter} \mathrm{of} \mathrm{the} \mathrm{circle} {\mathrm{x}}^2 + {\mathrm{y}}^2 = 9. \mathrm{If} \mathrm{\alpha } \mathrm{and} \mathrm{\beta } \mathrm{are} \mathrm{the} \mathrm{lengths} \mathrm{of} \mathrm{the} \mathrm{perpendiculars} \mathrm{from} \\ \mathrm{P} \mathrm{and} \mathrm{Q} \mathrm{on} \mathrm{the} \mathrm{straight} \mathrm{line}, \mathrm{x} + \mathrm{y} = 2 \mathrm{respectively}, \mathrm{then} \mathrm{the} \mathrm{maximum} \mathrm{value} \mathrm{of} \mathrm{\alpha \beta } \mathrm{is} ..... \end{gathered}$$

If the common tangent to the parabolas, y² = 4x and x² = 4y also touches the circle, x² + y² = c², then c is equal to :

• $\frac{1}{\sqrt{2}}$

• $\frac{1}{2\sqrt{2}}$

• $\frac{1}{2}$

• $\frac{1}{4}$

If the co–ordinates of two points A and B are $\left(\sqrt{7}, 0\right) \mathrm{and} \left( - \sqrt{7}, 0\right)$ respectively and P is any point on the conic, 9x² + 16y² = 144, then PA + PB is equal to :

• 8

• 16

• 9

• 6

If the line y = mx + c is a common tangent to the hyperbola $\frac{{\mathrm{x}}^2}{100} - \frac{{\mathrm{y}}^2}{64} = 1 \mathrm{and} \mathrm{the} \mathrm{circle} {\mathrm{x}}^2 + {\mathrm{y}}^2 = 36, \mathrm{then} \mathrm{which} \mathrm{one} \mathrm{of} \mathrm{the} \mathrm{following} \mathrm{is} \mathrm{true}$

• 4c² = 369

• 5m = 4

• c² = 369

• 8m + 5 = 0

If the length of the chord of the circle, x² + y² = r²(r > 0) along the line, y – 2x = 3 is r, then r² is equal to :

• $\frac{9}{5}$

• 12

• $\frac{12}{5}$

• $\frac{24}{5}$

Let L₁ be a tangent to the parabola y² = 4(x + 1) and L₂ be a tangent to the parabola y² = 8(x + 2) such that L₁ and L₂ intersect at right angles. Then L₁ and L₂ meet on the straight line :

• x + 3 = 0

• 2x + 1 = 0

• x + 2y = 0

• x + 2 = 0

Which of the following points lies on the locus of the foot of perpendicular drawn upon any tangent to the ellipse,  $\frac{{\mathrm{x}}^2}{4} + \frac{{\mathrm{y}}^2}{2} = 1$ from any of its foci?

• $\left( - 2, \sqrt{3}\right)$

• $\left( - 1, \sqrt{2}\right)$

• (1, 2)

• $\left( - 1, \sqrt{3}\right)$

If the normal at an end of latus rectum of an ellipse passes through an extremity of the minor axis, the eccentricity e of the ellipse satisfies : 

• ${\mathrm{e}}^4 + 2{\mathrm{e}}^2 - 1 = 0$

• ${\mathrm{e}}^2 + \mathrm{e} - 1$

• ${\mathrm{e}}^2 + 2\mathrm{e} - 1 = 0$

• ${\mathrm{e}}^4 + {\mathrm{e}}^2 - 1 = 0$

630.

The centre of the circle passing through the point (0, 1) and touching the parabola y = x² at the point (2, 4) is:

• $\left(\frac{3}{10}, \frac{{\displaystyle 16}}{5}\right)$

• $\left(\frac{- 53}{10}, \frac{{\displaystyle 16}}{5}\right)$

• $\left(\frac{ - 16}{5}, \frac{{\displaystyle 53}}{10}\right)$

• $\left(\frac{6}{5}, \frac{{\displaystyle 53}}{10}\right)$