If O is any point on the axis of a parabola and POP’ be any chord passing through O, and if PM and PʹMʹ be the ordinates of P and Pʹ, prove that AM . AMʹ =AO2 and PM . PʹMʹ = −4a . AO, where A is the vertex.
My Self Assessment
What is the equation to the chord of the parabola y2 = 8x which is bisected at the point (2, −3).
The general equation to a system of parallel chords in the parabola is 4x – y + k = 0. What is the equation to the
corresponding diameter?
If the tangents of a parabola at P and Q meet at T, prove that ST2 = SP . SQ, where S is the focus.
Prove that all circles on focal radii of a parabola as diameter touch the tangent at the vertex.
A circle and parabola intersect in four points; show that the algebraic sum of the ordinates of the four points is zero.
Show that the line joining one pair of these points and the line joining the other pair are equally inclined to the axis.
Find the locus of the point O when the three normals to the parabola y2 = 4ax from it are such that two of them make complementary angles with the axis.
Find the locus of the point O when the three normals to the parabola y2 = 4ax from it are such that two of them make angles with the axis the product of whose tangents is 2.
Find the locus of the point O when the three normals to the parabola y2 = 4ax from it are such that the sum of three angles made by them with the axis is constant (= λ).
The normals at three points P, Q and R of the parabola y2 = 4ax meet in a point O whose coordinates are h and k, prove that the centroid of triangle PQR lies on the axis.
The normals at three points P, Q and R of the parabola y2 = 4ax meet in a point O whose coordinates are h and k, prove that the sum of intercepts which the normals cut off from the axis is 2 (h + a).
A circle is described whose centre (0, 0) is the vertex and whose diameter is three quarters of the latus rectum of the parabola y2 = 4ax. Prove that the common chord of the circle and parabola bisects the distance between the vertex and the focus.
Trace the curve 4x2 – 4xy + y2 – 12x + 6y + 9 = 0
The area of the region bounded by the curves and the X-axis is
a) 1
b) 2
c) 3
d) 4
Find the equation of the circle which passes through the points (1, 0), (0, −6) and (3, 4).
ABCD is a square whose side is a; taking AB and AD as axes, prove that the equation of the circle circumscribing the square is x2 + y2 = a (x + y)
Find the equation to the circle which passes through the origin and cuts off intercepts equal to 3 and 4 from the axes.
Find the equation to the circle which passes through the origin and cuts off intercepts equal to h and k from the positive parts of the axes.
Find the equation of the circle of radius a which passes through two points on the axes of x which are at a distance b from the origin.
Find the equation of the circle which touches the Y-axis at the origin and passes through the point (b, c)
Find the equation to the tangent to the circle x2 + y2 – 6x + 4y = 12 which is parallel to the straight line 4x + 3y + 5 = 0.
Find the equation to the tangent to the circle x2 + y2 – 3x + 10y = 15 at the point (4, −11).
Find the equation to the tangent to the circle x2 + y2 + 2gx + 2fy + c = 0 which is parallel to the line x + 2y – 6 = 0.
Prove that the straight line touches the circle x2 + y2 = c2 and find the point of contact.
