The straight lines x + y = 0, 5x + y = 4 and x + 5y = 4 form

• an isosceles triangle

• an equilateral triangle

• a scalene triangle

• a right angled triangle

If z = x + iy, where x and y are real numbers and i = $\sqrt{- 1}$, then the points (x, y) for which $\frac{\mathrm{z} - 1}{\mathrm{z} - \mathrm{i}}$ is real, lie on

• an ellipse

• a circle

•  a parabola

• a straight line

The equation 2x² + 5xy - 12y² = 0 represents a

• circle

• pair of non-perpendicular intersecting straight lines

• pair of perpendicular straight lines

• hyperbola

The number oflines which pass through the point (2, - 3) and are at a distance 8 from the point (- 1, 2) is

• infinite

• 4

• 2

• 0

A line passing through the point of intersection of x + y = 4 and x - y = 2 makes an angle ${\tan }^{-1}\left(\frac{3}{4}\right)$ with the x-axis. It intersects the parabola y² = 4(x - 3) at points (x₁, y₁) and (x₂, y₂) respectively. Then, $\left|{\mathrm{x}}_1 - {\mathrm{x}}_2\right|$ is equal to

• $\frac{16}{9}$

• $\frac{32}{9}$

• $\frac{40}{9}$

• $\frac{80}{9}$

The line joinining $\mathrm{A}\left(\mathrm{bcos}\left(\mathrm{\alpha }\right), \mathrm{bsin}\left(\mathrm{\alpha }\right)\right)$ and $\mathrm{B}\left(\mathrm{acos}\left(\mathrm{\beta }\right), \mathrm{asin}\left(\mathrm{\beta }\right)\right)$, where a $\ne$ b, is produced to the point M(x, y) so that AM : MB = b : a. Then, $\mathrm{xcos}\left(\frac{\mathrm{\alpha } + \mathrm{\beta }}{2}\right) + \mathrm{ysin}\left(\frac{\mathrm{\alpha } + \mathrm{\beta }}{2}\right)$ is equal to

• 0

• 1

• - 1

• a² + b²

47.

The number of integer values of m, for which the x - coordinate of the point of intersection of the lines 3x + 4y= 9 and y =mx + 1 is also an integer, is

• 0

• 2

• 4

• 1

If a straight line passes through the point ($\mathrm{\alpha }, \mathrm{\beta }$) and the portion of the line intercepted between the axes is divided equally at that point, then $\frac{\mathrm{x}}{\mathrm{\alpha }} + \frac{\mathrm{y}}{\mathrm{\beta }}$ is

• 0

• 1

• 2

• 4

A straight line through the point of intersection of the lines x + 2y = 4 and 2x + y = 4 meets the coordinate axes at A and B. The locus of the mid-point of AB is

• 3(x + y) = 2xy

• 2(x + y) = 3xy

• 2(x + y) = xy

• x + y = 3xy

Find the image of the point (- 8, 12) with respect to the line 4x + 7y + 13 = 0.