Prove that the locus of the middle point of all chords of parabola y2 = 4ax which are drawn through the vertex is the parabola y2 = 2ax.
My Self Assessment
Find the length of the common chord of the circles, whose equations are (x – a)2 + y2 = a2 x2 + (y – b)2 = b2
Prove that the equation to the circle whose diameter is this common chord is (a2 + b2) (x2 + y2) = 2ab (bx + ay)
Prove that the length of the common chord of the circles whose equations are (x – a)2 + (y – b)2 = c2 and (x – b)2 + (y – a)2 = c2 is
Prove that the following pair of circles intersect orthogonally x2 + y2 – 2ax + c = 0 and x2 + y2 + 2by – c = 0
Find the radical axis of the pair of circles x2 + y2 – 3x – 4y + 5 = 0 and 3x2 + 3y2 – 7x + 8y + 11 = 0
Find the radical axis of the set of circles (x – 2)2 + (y – 3)2 = 36 (x + 3)2 + (y + 2)2 = 43 (x – 4)2 + (y + 5)2 =64
Find the equation to the straight lines joining the origin to the points of intersection of x2 + y2 – 4x – 2y = 4 and x2 + y2 – 2x – 4y – 4 = 0
Find the equation to the circle which cuts orthogonally each of the three circles x2 + y2 + 2x + 17y + 4 = 0 x2 + y2 + 7x + 6y + 11 = 0 x2 + y2 – x +22y + 3 = 0
Find the equation to the circle cutting orthogonally the three circles x2 + y2 = a2 (x – c)2 + y2 = a2 x2 + (y – b)2 = a2
Obtain the equation of the parabola whose focus is the point (2, 3) and whose directrix is the line x – 4y + 3 = 0
For what point of the parabola y2 = 18x is the ordinate equal to three times the abscissa?
Prove that the equation to the parabola whose vertex and focus are on the axis of x at a distance a and a' from the origin respectively is y2 = 4 (a' – a) (x – a)
In the parabola y2 = 6x, find the equation to the chord through the vertex and the negative end of the latus rectum.
In the parabola y2 = 6x, find the equation to any chord through the point on the curve whose abscissa is 24.
PQ is a double ordinate of a parabola. Find the locus of its point of trisection.
A double ordinate of the curve y2 = 4px is of length 8p; prove that the lines from the vertex to the two ends are at right angles.
Find the equation to the tangent to the parabola y2 = 7x which is parallel to the straight line 4y – x + 3 = 0. Find also its point of contact.
A tangent to the parabola y2 = 4ax makes an angle of 60° with the X – axis, find its point of contact.
Find points of the parabola y2 = 4ax at which the tangent is inclined at 30° to the axis.
Find points of the parabola y2 = 4ax at which the normal is inclined at 30° to the axis.
Find the equation to the tangents to the parabola y2 = 9x, which goes through the point (4, 10).
Prove that x + y = 1 touches the parabola y = x – x2.
Prove that the straight line y = mx + c touches the parabola y2 = 4a (x + a) if
Prove that the straight line lx + my + n = 0 touches the parabola y2 = 4ax if ln = am2
