Multiple Choice QuestionsIf the circles x2 + y2 + 2ax + cy + a = 0 and x2 + y2 – 3ax + dy – 1 = 0 intersect in two distinct points P and Q then the line 5x + by – a = 0 passes through P and Q for
exactly one value of a
no value of a
infinitely many values of a
infinitely many values of a
A circle touches the x-axis and also touches the circle with centre at (0, 3) and radius 2. The locus of the centre of the circle is
an ellipse
a circle
a hyperbola
a hyperbola
If a circle passes through the point (a, b) and cuts the circle x2 + y2 = p2 orthogonally, then the equation of the locus of its centre is
x2 + y2 – 3ax – 4by + (a2 + b2 – p2 ) = 0
2ax + 2by – (a2 – b2 + p2 ) = 0
x2 + y2 – 2ax – 3by + (a2 – b2 – p2 ) = 0
x2 + y2 – 2ax – 3by + (a2 – b2 – p2 ) = 0
D.
x2 + y2 – 2ax – 3by + (a2 – b2 – p2 ) = 0
Let the centre be (α, β)
∵It cut the circle x2 + y2 = p2 orthogonally
2(-α) × 0 + 2(-β) × 0 = c1 – p2
c1 = p2 Let equation of circle is x2 + y2 - 2αx - 2βy + p2 = 0
It pass through (a, b) ⇒ a2 + b2 - 2αa - 2βb + p2 = 0
Locus ∴ 2ax + 2by – (a2 + b2 + p2 ) = 0.
An ellipse has OB as semi-minor axis, F and F′ its focii and the angle FBF′ is a right angle. Then the eccentricity of the ellipse is
1/2
1/4
1/4
If the pair of lines ax2 + 2(a + b)xy + by2 = 0 lie along diameters of a circle and divide the circle into four sectors such that the area of one of the sectors is thrice the area of another sector then
3a2 – 10ab + 3b2 = 0
3a2 – 2ab + 3b2 = 0
3a2 + 10ab + 3b2 = 0
3a2 + 10ab + 3b2 = 0
If a circle passes through the point (a, b) and cuts the circle x2 +y2= 4 orthogonally, then the locus of its centre is
2ax +2by + (a2 +b2+4)=0
2ax +2by - (a2 +b2+4)=0
2ax -2by - (a2 +b2+4)=0
2ax -2by - (a2 +b2+4)=0
A variable circle passes through the fixed point A (p, q) and touches x-axis. The locus of the other end of the diameter through A is
(x-p)2 = 4qy
(x-q)2 = 4py
(y-p)2 = 4qx
(y-p)2 = 4qx
If the lines 2x + 3y + 1 = 0 and 3x – y – 4 = 0 lie along diameters of a circle of circumference 10π, then the equation of the circle is
x2 + y2- 2x +2y -23 = 0
x2 - y2- 2x -2y -23 = 0
x2 - y2- 2x -2y +23 = 0
x2 - y2- 2x -2y +23 = 0
If a ≠ 0 and the line 2bx + 3cy + 4d = 0 passes through the points of intersection of the parabolas y2+ 4ax = and x2+ 4ay = , then
d2 + (2b+3c)2 = 0
d2 +(3d+2c2) = 0
d2 + (2b-3c)2 = 0
d2 + (2b-3c)2 = 0
Switch
