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)

• $\left( - 1, \sqrt{3}\right)$

436.

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 : 

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

• ${\mathrm{e}}^2 + \mathrm{e} - 1$

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

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

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

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

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

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

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

The radius of the larger circle lying in the first quadrant and touching the line 4x + 3y - 12 =0 and the co-ordinate axes,is

• 5

• 6

• 7

• 8

The parabola with directrix x + 2y - 1 = 0 and focus (1, 0) is

• 4x² - 4xy + y² - 8x + 4y + 4 = 0
  
  

• 4x² + 4xy + y² - 8x + 4y + 4 = 0
  
  

• 4x² + 5xy + y² + 8x - 4y + 4 = 0
  
  

• 4x - 4xy + y - 8x - 4y + 4 = 0

The length of the common chord of the circles of radii 15 and 20, whose centres are 25 unit of distance apart, is

• 12

• 16

• 24

• 25