The equation of a circle whose end points of a diameter are (x₁, y₁) and x₂, y₂) is

• $\left(\mathrm{x} - {\mathrm{x}}_1\right)\left(\mathrm{x} - {\mathrm{x}}_2\right) + \left(\mathrm{y} - {\mathrm{y}}_1\right)\left(\mathrm{y} - {\mathrm{y}}_2\right) = {\mathrm{x}}^2 + {\mathrm{y}}^2$

• ${\left(\mathrm{x} - {\mathrm{x}}_1\right)}^2 + {\left(\mathrm{y} - {\mathrm{y}}_1\right)}^2 = {\mathrm{x}}_2{\mathrm{y}}_2$

• ${\mathrm{x}}^2 + {\mathrm{y}}^2 + 2{\mathrm{x}}_1{\mathrm{x}}_2 + 2{\mathrm{y}}_1{\mathrm{y}}_2 = 0$

• $\left(\mathrm{x} - {\mathrm{x}}_1\right)\left(\mathrm{x} - {\mathrm{x}}_2\right) + \left(\mathrm{y} - {\mathrm{y}}_1\right)\left(\mathrm{y} - {\mathrm{y}}_2\right) = 0$

The second degree equation x² + 4y - 2x - 4y + 2 = 0 represents

• A point

• An ellipse of semi-major axis 1

• $\mathrm{An} \mathrm{ellipse} \mathrm{with} \mathrm{eccentricity} \frac{\sqrt{3}}{2}$

• None of the above

If y-axis touches the circle x² + y² + gx + fy + $\frac{\mathrm{c}}{4}$ = 0, then the normal at this point intersects the circle at the point

• $\left(- \frac{\mathrm{g}}{2}, - \frac{\mathrm{f}}{2}\right)$

• $\left( - \mathrm{g}, - \frac{\mathrm{f}}{2}\right)$

• $\left(- \frac{\mathrm{g}}{2}, \mathrm{f}\right)$

• ( - g, - f)

A flower-bed in the form of a sector has been fenced by a wire of 40 m length. If the flower-bed has the greatest possible area, then what is the radius of the sector ?

• 25m

• 20m

• 10m

• 5m

The two circles x² + y² = r² and x² + y² - 10x + 16 = 0 intersect at two distinct points. Then which one of the following is correct ? 

• $2 < \mathrm{r} < 8$

• r = 2 or r = 8

• $\mathrm{r} < 2$

• $\mathrm{r} > 2$

What is the equation of the circle which passes through the points (3, - 2) and (- 2, 0) and having its centre on the line 2x - y - 3 = 0 ?

• ${\mathrm{x}}^2 + {\mathrm{y}}^2 + 3\mathrm{x} + 2 = 0$

• ${\mathrm{x}}^2 + {\mathrm{y}}^2 +\hspace{0.17em}3\mathrm{x} + 12\mathrm{y} + 2 = 0$

• ${\mathrm{x}}^2 + {\mathrm{y}}^2 + 2\mathrm{x} = 0$

• x${\mathrm{x}}^2 + {\mathrm{y}}^2 = 5$

What is the equation of the ellipse having foci (± 2, 0) and the eccentricity $\frac{1}{4} ?$

• $\frac{{\mathrm{x}}^2}{64} + \frac{{\mathrm{y}}^2}{60} = 1$

• $\frac{{\mathrm{x}}^2}{60} + \frac{{\mathrm{y}}^2}{64} = 1$

• $\frac{{\mathrm{x}}^2}{20} + \frac{{\mathrm{y}}^2}{24} = 1$

• $\frac{{\mathrm{x}}^2}{24} + \frac{{\mathrm{y}}^2}{20} = 1$

What is the maximum area of a triangle that can be inscribed in a circle of radius a ?

• $\frac{3{\mathrm{a}}^2}{4}$

• $\frac{{\mathrm{a}}^2}{2}$

• $\frac{3\sqrt{3}{\mathrm{a}}^2}{4}$

• $\frac{\sqrt{3}{\mathrm{a}}^2}{4}$

19.

A man running round a race course notes that the sum of the distances of two flag-posts from him is always 10 m and the distance between the flag-posts is 8 m. The area of the path he encloses is :

• 18$\mathrm{\pi }$ square metres

• 15$\mathrm{\pi }$ square metres

• 12$\mathrm{\pi }$ square metres

• $8\mathrm{\pi } \mathrm{square} \mathrm{metres}$

The position of the point (1, 2) relative to the ellipse 2k² + 7 = 20 is : 

• Outside the ellipse

• Inside the ellipse but not at the focus

• On the ellipse

• at the focus