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. The eccentricity of the hyperbola $9y^2 - 4x^2 = 36$ is
2. The equation of the circle with centre $(h, k)$ and touching the x-axis is
3. If the parabola $y^2 = 4ax$ passes through the point $(3, 2)$, then the length of its latus rectum is
4. The equation of the ellipse whose foci are $(\pm 2, 0)$ and eccentricity is $1/2$ is
5. The line $x + y = 2$ touches the circle $x^2 + y^2 = 2$ at the point
6. Find the equation of the circle which passes through the points $(2, -2)$ and $(3, 4)$ and whose centre lies on the line $x + y = 2$.
7. Find the coordinates of a point on the parabola $y^2 = 8x$ whose focal distance is 4.
8. Find the equation of the hyperbola with foci $(0, \pm 3)$ and vertices $(0, \pm \frac{\sqrt{11}}{2})$.
9. Find the equation of the circle passing through the origin and making intercepts whose lengths are 3 and 4 on the positive x and y axes respectively.
10. An equilateral triangle is inscribed in the parabola $y^2 = 4ax$, where one vertex is at the vertex of the parabola. Find the length of the side of the triangle.
11. A rod of length 15 cm rests in between two coordinate axes in such a way that the end point A lies on x-axis and end point B lies on y-axis. A point P(x, y) is taken on the rod in such a way that $AP = 6$ cm. Show that the locus of P is an ellipse.
12. Case Study: An arch is in the form of a semi-ellipse. It is 12 m wide and 4 m high at the center. The height of the arch changes as we move away from the center.
    (i) Find the equation of the ellipse (taking center as origin).
    (ii) Find the eccentricity of the ellipse.
    (iii) Find the height of the arch at a point 2 m from the center.