Straight Lines

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 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.

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