The locus of the mid-point of the line segment joining the focus to a moving point on the parabola y2 = 4ax is another parabola with directrix
x = 0
y2 = 4ax … (1)
Its focus is S(a, 0). Any point on (1) is P (t12, 2t1).
Let M(x, y) be mid-point of chord PS, then
Eliminating t between (2) and (3).
Its directrix is X = –A .
Let a and b be non-zero real numbers. Then the equation (ax2 + by2 + c)(x2 – 5xy + 6y2) = 0 represents
The pints of intersection of the two ellipse x2 + 2y2 – 6x – 12y + 23 = 0 and 4x2 + 2y2 – 20x – 12y + 35 = 0.
The tangent at any point P of the hyperbola makes an intercept of length p between the point of contact and the transverse axis of the hyperbola, p1, p2 are the lengths of the perpendiculars drawn from the foci on the normal at P, then
The length of chord of contact of the tangents drawn from the point (2, 5) to the parabola y2 = 8x, is
The locus of a point whose some of the distance from the origin and the line x = 2 is 4 units, is
The length of the subnormal to the parabola y2 = 4ax at any point is equal to
The slope of the normal at the point (at2, 2at) of parabola y2 = 4ax is
Equation of locus of a point whose distance from point (a, 0) is equal to its distance from y-axis is
Through the vertex O of parabola y2 = 4x, chords OP and OQ are drawn at right angles to one another. The locus of the middle point of PQ is
The equation of common tangent to the curves y2 = 8x and xy = –1 is