∆ PRA, sin θ = Since cos θ + sin θ = or Thus the locus of P is an ellipse. Miscellaneous Exercise on Chapter . If a parabolic reflector is cm in diameter and cm deep, find the focus. .
An arch is in the form of a parabola with its axis vertical. The arch is m high and m wide at the base. How wide is it m from the vertex of the parabola? .
The cable of a uniformly loaded suspension bridge hangs in the form of a parabola. The roadway which is horizontal and m long is supported by vertical wires attached to the cable, the longest wire being m and the shortest being m. Find the length of a supporting wire attached to the roadway m from the middle. .
An arch is in the form of a semi-ellipse. It is m wide and m high at the centre. Find the height of the arch at a point . m from one end.
. A rod of length cm moves with its ends always touching the coordinate axes. Determine the equation of the locus of a point P on the rod, which is cm from the end in contact with the x -axis. .
Find the area of the triangle formed by the lines joining the vertex of the parabola x = y to the ends of its latus rectum. . A man running a racecourse notes that the sum of the distances from the two flag posts from him is always m and the distance between the flag posts is m. Find the equation of the posts traced by the man.
. An equilateral triangle is inscribed in the parabola y = ax , where one vertex is at the vertex of the parabola. Find the length of the side of the triangle. CONIC SECTIONS Summary In this Chapter the following concepts and generalisations are studied.
® A circle is the set of all points in a plane that are equidistant from a fixed point in the plane. ® The equation of a circle with centre ( h , k ) and the radius r is ( x – h ) + ( y – k ) = r . ® A parabola is the set of all points in a plane that are equidistant from a fixed line and a fixed point in the plane. ® The equation of the parabola with focus at ( a , ) a > and directrix x = – a is y = ax .