The value of for which the curve (7x + 5)2 + (7y + 3)2 = (4x + 3y - 24)2 represents a parabola is
The equation of the common tangent with positive slope to the parabola y2 = 8x and the hyperbola 4x2 - y2 = 4 is
The point on the parabola y2 = 64x which is nearest to the line 4x + 3y + 35 = 0 has coordinates
(9, - 24)
(1, 81)
(4, - 16)
(- 9, - 24)
Let z1, z2 be two fixed complex numbers in the argand plane and z be an arbitrary point satisfying Then, the locus of z will be
an ellipse
a straight line joining z1 and z2
a parabola
a bisector of the line segment joining z1 and z2
Let z, be a fixed point on the circle of radius 1 centered at the origin in the Argand plane and . Consider an equilateral triangle inscribed in the circle with z1, z2, z3 as the vertices taken in the counterclockwise direction. Then, z1z2z3 is equal to
z12
z13
z14
z1
If the circle x2 + y2 + 2gx + 2fy + c = 0 cuts the three circles x2 + y2 - 5 = 0, x2 + y2 - 8x - 6y + 10 = 0 and x2 + y2 - 4x + 2y - 2 = 0 at the extremities of their diameters, then
c = - 5
fg = 147/25
g + 2f = c + 2
4f = 3g
Lines x + y = 1 and 3y = x + 3 intersect the ellipse x2 + 9y2 = 9 at the points P,Q and R. The area of the PQR is
For the variable , the locus of the point of intersection of the lines 3tx - 2y + 6t = 0 and 3x + 2ty - 6 = 0 is
A.
Given equation of lines are
3tx - 2y + 6t = 0 ...(i)
and 3x + 2ty - 6 = 0 ...(ii)
On multiplying Eq. (i) by t and then adding in Eq. (ii), we get
(3t2 + 3)x + 6t2 - 6 = 0
On multiplying Eq. (ii) by t and then subtract from Eq. (i), we get
, which represents an ellipse.
The locus of the mid-points of the chords of an ellipse x2 + 4y2 = 4 that are drawn from the positive end of the minor axis, is
a circle with centre and radius 1
a parabola with focus and directrix x = - 1
an ellipse with centre , major axis and minor axis
a hyperbola with centre , transverse axis 1 and conjugate axis