The equation of the ellipse whose centre is at origin, major axis is along x-axis with eccentricity, $\frac{3}{4}$ and latus rectum 4 units is :

• $\frac{{\mathrm{x}}^2}{1024} + \frac{7{\mathrm{y}}^2}{64} = 1$

• $\frac{49{\mathrm{x}}^2}{1024} + \frac{7{\mathrm{y}}^2}{64} = 1$

• $\frac{7{\mathrm{x}}^2}{1024} + \frac{49{\mathrm{y}}^2}{64} = 1$

• $\frac{{\mathrm{x}}^2}{1024} + \frac{{\mathrm{y}}^2}{64} = 1$

The equation of the circle which passes through the points (1, 0), (0, - 6) and (3, 4) is :

• $4{\mathrm{x}}^2 + 4{\mathrm{y}}^2 + 142\mathrm{x} + 47\mathrm{y} + 140 = 0$

• $4{\mathrm{x}}^2 +\hspace{0.17em}4{\mathrm{y}}^2 - 142\mathrm{x} - 47\mathrm{y} + 138 = 0$

• $4{\mathrm{x}}^2 +\hspace{0.17em}4{\mathrm{y}}^2 - 142\mathrm{x} + 47\mathrm{y} + 138 = 0$

• $4{\mathrm{x}}^2 +\hspace{0.17em}4{\mathrm{y}}^2 + 150\mathrm{x} - 49\mathrm{y} + 138 = 0$

23.

A cylindrical jar without a lid has to be constructed using a given surface area of a metal sheet. If the capacity of the jar is to be maximum, then the diameter of the jar must be k times the height of the jar. The value of k is

• 1

• 2

• 3

• 4

Geometrically $\mathrm{Re}\left({\mathrm{z}}^2 - \mathrm{i}\right) = 2, \mathrm{where} \mathrm{i} = \sqrt{ - 1} \mathrm{and} \mathrm{Re} \mathrm{is} \mathrm{the} \mathrm{real} \mathrm{part}, \mathrm{represents}$

• circle

• ellipse

• rectangular hyperbola

• parabola

The following table gives the monthly expenditure of two families :

In constructing a pie diagram to the above data, the radii of the circles are to be choosen by which one of the following ratios ?

• 1 : 1

• 10 : 9

• 100 : 91

• 5 : 4

If a circle of radius b units with centre at (0, b) touches the line y = x $- \sqrt{2}$, then what is the value of b ?

• $2 + \sqrt{2}$

• $2 - \sqrt{2}$

• $2\sqrt{2}$

• $\sqrt{2}$

Consider the two circles

$$\begin{gathered} {\left(\mathrm{x} - 1\right)}^2 + {\left(\mathrm{y} - 3\right)}^2 = {\mathrm{r}}^2 \mathrm{and} \\ {\mathrm{x}}^2 + {\mathrm{y}}^2 - 8\mathrm{x} + 2\mathrm{y} + 8 = 0 \end{gathered}$$

What is the distance between the centres of the two circles ?

• 5 units

• 6 units

• 8 units

• 10 units

Consider the two circles

$$\begin{gathered} {\left(\mathrm{x} - 1\right)}^2 + {\left(\mathrm{y} - 3\right)}^2 = {\mathrm{r}}^2 \mathrm{and} \\ {\mathrm{x}}^2 + {\mathrm{y}}^2 - 8\mathrm{x} + 2\mathrm{y} + 8 = 0 \end{gathered}$$

If circles intersect at two distinct. points, then which one of the following is correct ?

• r = 1

• $1 < \mathrm{r} < 2$

• r = 2

• $2 < \mathrm{r} < 8$

Consider a circle passing through the origin and the points (a, b) and (- b, - a). On which line does the centre of the circle lie ?

• x + y = 0

• x - y = 0

• x + y = a + b

• x - y = a² - b²

Consider a circle passing through the origin and the points (a, b) and (- b, - a). What is the sum of the squares of the intercepts cut off by the circle on the axes ?

• ${\left(\frac{{\mathrm{a}}^2 + {\mathrm{b}}^2}{{\mathrm{a}}^2 - {\mathrm{b}}^2}\right)}^2$

• $2{\left(\frac{{\mathrm{a}}^2 + {\mathrm{b}}^2}{\mathrm{a} - \mathrm{b}}\right)}^2$

• $4{\left(\frac{{\mathrm{a}}^2 + {\mathrm{b}}^2}{\mathrm{a} - \mathrm{b}}\right)}^2$

• None of the above