51.

The coordinates of the two points lying on x + y = 4 and at a unit distance from the straight line 4x + 3y = 10 are

• (- 3, 1), (7, 11)

• (3, 1), (- 7, 11)

• (3, 1), (7, 11)

• (5, 3), (- 1, 2)

The equation of the locus of the point of intersection of the straight lines
$$\begin{gathered} \mathrm{xsin}\left(\mathrm{\theta }\right) + (1 - \cos \left(\mathrm{\theta }\right))\mathrm{y} = \mathrm{a} \sin \left(\mathrm{\theta }\right) \mathrm{and} \\ \mathrm{xsin}\left(\mathrm{\theta }\right) - (1 + \cos \left(\mathrm{\theta }\right)) \mathrm{y} + \mathrm{a} \sin \left(\mathrm{\theta }\right)= 0 \mathrm{is} \end{gathered}$$

• $\mathrm{y} = \pm \mathrm{ax}$

• $\mathrm{x} = \pm \mathrm{ay}$

• y² = 4x

• x² + y² = a²

The straight line 3x + y divides the line segment joining the points (1, 3) and (2, 7) in the ratio

• 3 : 4 externally

• 3 : 4 internally

• 4 : 5 internally

• 5 : 6 externally

If the sum of distances from a point P on two mutually perpendicular straight lines is 1 unit, then the locus of P is

• a parabola

• a circle

• an ellipse

• a straight line

The straight line x + y - 1 = 0 meets the circle x² + y² - 6x - 8y = 0 at A and B. Then the equation of the circle of which AB is a diameter is

• x² + y² - 2y - 6 = 0

• x² + y² + 2y - 6 = 0

• 2(x² + y²) + 2y - 6

• 3(x² + y²) + 2y - 6 = 0

The coordinates of the point on the curve y = x² - 3x + 2 where the tangent is perpendicular to the straight line y = x are 

• (0, 2)

• (1, 0)

• (- 1, 6)

• (2, - 2)

The equations of the lines through (1, 1) and making angles of 45° with the line x + y = 0 are

• x - 1 = 0, x - y = 0

• x - y = 0, y - 1 = 0

• x + y - 2 = 0, y - 1 = 0

• x - 1 = 0, y - 1 = 0

The number of points on the line x + y = 4 which are unit distance apart from the line 2x + 2y = 5 is

• 0

• 1

• 2

• $\infty$

The point (- 4, 5) is the vertex of a square and one of its diagonals is 7x - y + 8 = 0. The equation of the other diagonal is

• 7x - y + 23 = 0

• 7y + x = 30

• 7y + x = 31

• x - 7y = 30

A line through the point A(2, 0) which makes an angle of 30° with the positive direction of x-axis is rotated about A in clockwise direction through an angle 15°. Then, the equation of the straight line in the new position is

• $\left(2 - \sqrt{3}\right)\mathrm{x} + \mathrm{y} - 4 + 2\sqrt{3} = 0$

• $\left(2 - \sqrt{3}\right)\mathrm{x} - \mathrm{y} - 4 + 2\sqrt{3} = 0$

• $\left(2 - \sqrt{3}\right)\mathrm{x} - \mathrm{y} + 4 + 2\sqrt{3} = 0$

• $\left(2 - \sqrt{3}\right)\mathrm{x} + \mathrm{y} + 4 + 2\sqrt{3} = 0$