21.

If p and q are the perpendicular distances from the origin to the straight lines xsec$\left(\mathrm{\theta }\right)$ - ycosec($\left(\mathrm{\theta )}\right)$ = $\mathrm{\alpha }$ and xcos($\left(\mathrm{\theta }\right)$) + ysin($\left(\mathrm{\theta }\right)$) = $\mathrm{\alpha }$cos(2$\left(\mathrm{\theta )}\right)$, then

• 4p² + q² = a²

• p² + q² = a²

• p² + 2q² = a²

• 4p² + q² = 2a²

If 2x + 3y =5 is the perpendicular bisector of the line segment joining the points A (1, 1/3) and B, then B is equal to

• $\left(\frac{21}{13}, \frac{49}{39}\right)$

• $\left(\frac{17}{13}, \frac{31}{39}\right)$

• $\left(\frac{7}{13}, \frac{49}{39}\right)$

• $\left(\frac{21}{13}, \frac{31}{39}\right)$

If the points (1, 2) and (3, 4) lie on the same side of the straight line 3x - 5y + a = 0, then a lies in the set

• [7, 11]

• R - [7, 11]

• $\left(7, \infty \right)$

• $\left(- \infty , 11\right)$

The equation of the pair of lines passing through the orign whose sum and product of slopes are respectively the arithmetic mean and geometric mean of 4 and 9 is

• 12x² - 13xy + 2y² = 0

• 12x² + 13xy + 2y² = 0

• 12x² - 15xy + 2y² = 0

• 12x² + 15xy - 2y² = 0

The equation ${\mathrm{x}}^2 - 5\mathrm{xy} + {\mathrm{py}}^2 + 3\mathrm{x} - 8\mathrm{y} + 2 = 0$ represents a pair of straight lines. If $\mathrm{\theta }$ is the angle between them, then $\sin \left(\mathrm{\theta }\right)$ is equal to

• $\frac{1}{\sqrt{50}}$

• $\frac{1}{7}$

• $\frac{1}{5}$

• $\frac{1}{\sqrt{10}}$

If the equation ${\mathrm{ax}}^2 + 2\mathrm{hxy} + {\mathrm{by}}^2 +\hspace{0.17em}2\mathrm{gx} + 2\mathrm{fy} + \mathrm{c} = 0$represents a pair of, straight lines, then the square of the distance of their point of intersection from the origin is

• $\frac{\mathrm{c}\left(\mathrm{a} + \mathrm{b}\right) - {\mathrm{af}}^2 - {\mathrm{bg}}^2}{\mathrm{ab} - {\mathrm{h}}^2}$

• $\frac{\mathrm{c}\left(\mathrm{a} + \mathrm{b}\right) + {\mathrm{f}}^2 + {\mathrm{g}}^2}{\mathrm{ab} - {\mathrm{h}}^2}$

• $\frac{\mathrm{c}\left(\mathrm{a} + \mathrm{b}\right) - {\mathrm{f}}^2 - {\mathrm{g}}^2}{\mathrm{ab} - {\mathrm{h}}^2}$

• $\frac{\mathrm{c}\left(\mathrm{a} + \mathrm{b}\right) - {\mathrm{f}}^2 - {\mathrm{g}}^2}{{\left(\mathrm{ab} - {\mathrm{h}}^2\right)}^2}$

The circle $4{\mathrm{x}}^2 + 4{\mathrm{y}}^2 - 12\mathrm{x} - 12\mathrm{y} + 9 = 0$

• Touches both the axes

• Touches the x-axis only

• Touches the y-axis only

• Does not touch the axes

If the length of the tangent from (h, k) to the circle x² + y² = 16 is twice the length of the tangent from the same point to the circle x² + y² + 2x + 2y = 0, then

• ${\mathrm{h}}^2 + {\mathrm{k}}^2 + 4\mathrm{h} + 4\mathrm{k} + 16 = 0$

• ${\mathrm{h}}^2 + {\mathrm{k}}^2 + 3\mathrm{h} + 3\mathrm{k} = 0$

• $3{\mathrm{h}}^2 + 3{\mathrm{k}}^2 + 8\mathrm{h} + 8\mathrm{k} + 16 = 0$

• $3{\mathrm{h}}^2 + 3{\mathrm{k}}^2 + 4\mathrm{h} + 4\mathrm{k} + 16 = 0$

($\mathrm{\alpha }$, 0) and (b, 0) are centres of two circles belonging to a coaxial system of which y-axis is the radical axis. If radius of one of the circles is 'r', then the radius of the other circle is

• ${\left({\mathrm{r}}^2 + {\mathrm{b}}^2 + {\mathrm{a}}^2\right)}^{\raisebox{1ex}{$1$}\!\left/ \!\raisebox{-1ex}{$2$}\right.}$

• ${\left({\mathrm{r}}^2 + {\mathrm{b}}^2 - {\mathrm{a}}^2\right)}^{\raisebox{1ex}{$1$}\!\left/ \!\raisebox{-1ex}{$2$}\right.}$

• ${\left({\mathrm{r}}^2 + {\mathrm{b}}^2 - {\mathrm{a}}^2\right)}^{\raisebox{1ex}{$1$}\!\left/ \!\raisebox{-1ex}{$3$}\right.}$

• ${\left({\mathrm{r}}^2 + {\mathrm{b}}^2 + {\mathrm{a}}^2\right)}^{\raisebox{1ex}{$1$}\!\left/ \!\raisebox{-1ex}{$3$}\right.}$

If the circle x² + y² + 4x - 6y + c =0 bisects the circumference of the circle x² + y² - 6x + 4y - 12 = 0, then c is equal to

• 16

• 24

• - 42

• - 62