Questions covered in Hard

Review the question prompts below before starting the timed test. Try each problem independently, then use the interactive test to record your answers and review the worked solutions.

1. If the lines $2x + y - 3 = 0$, $5x + ky - 3 = 0$ and $3x - y - 2 = 0$ are concurrent, then the value of $k$ is
2. The equation of the line passing through $(1, 2)$ and making an angle of $30^\circ$ with the y-axis is
3. If $p$ is the length of perpendicular from the origin on the line $\frac{x}{a} + \frac{y}{b} = 1$, then
4. The coordinates of the orthocentre of the triangle formed by the lines $x = 0, y = 0$ and $x + y = 1$ is
5. The area of the triangle formed by the lines $y = x, y = 2x$ and $y = 3x + 4$ is
6. Find the equation of the line passing through the point $(2, 2)$ such that the sum of its intercepts on the axes is 9.
7. Find the equation of the line mid-parallel to the lines $2x - 5y + 7 = 0$ and $2x - 5y - 3 = 0$.
8. Find the angle between the lines $y - \sqrt{3}x - 5 = 0$ and $\sqrt{3}y - x + 6 = 0$.
9. A ray of light passing through the point $(1, 2)$ reflects on the x-axis at point A and the reflected ray passes through the point $(5, 3)$. Find the coordinates of A.
10. Find the equation of the line passing through the intersection of the lines $4x + 7y - 3 = 0$ and $2x - 3y + 1 = 0$ that has equal intercepts on the axes.
11. If $p$ and $q$ are the lengths of perpendiculars from the origin to the lines $x \cos \theta - y \sin \theta = k \cos 2\theta$ and $x \sec \theta + y \csc \theta = k$, respectively, prove that $p^2 + 4q^2 = k^2$.
12. Case Study: A person is standing at the junction (crossing) of two straight paths represented by the equations $2x - 3y + 4 = 0$ and $3x + 4y - 5 = 0$. He wants to reach the path whose equation is $6x - 7y + 8 = 0$ in the least time.
    (i) Find the position of the person (intersection point).
    (ii) Find the equation of the path he should follow.
    (iii) Find the minimum distance he has to walk.