The range of parameter `' a '`
for which the variable line `y=2x+a`
lies between the circles `x^2+y^2-2x-2y+1=0`
and `x^2+y^2-16 x-2y+61=0`
without intersecting or touching either circle is
(a)`a in (2sqrt(5)-15 ,0)`
(b) `a in (-oo, 2sqrt(5)-15,)`
(c)`a in (2sqrt(5)-15,-sqrt(5)-1)`
(d) `a in (-sqrt(5)-1,oo)`
Text Solution
Verified by Experts
The given circles are `C_(1) : (x-1)^(1)+(y-1)^(2)=1,r_(1)=1` and `C_(2) : (x-8)^(2)+(y-1)^(2)=4,r_(2)=2` The line `y-2x-a=0` will lie between these circles if the centers of the circles lie on opposite sides of the line, i.e., `(1-2-a)(1-16-a)lt0` or `a in (-15,-1)` The line will not touch or intersect the circles if `(|1-2-a|)/(sqrt(5))gt1,(|1-16-1|)/(sqrt(5))gt2` or `|1+a| gt sqrt(5),|15+a| gt 2sqrt(5)` `i.e., `agt sqrt(5)-1` or `alt sqrt(5) -1` and `agt 2 sqrt(5)-15` or `alt - 2 sqrt(5)-15` Hence, the common values of a are `(2 sqrt(5)-15,-sqrt(5)-1)`.
Find the obtuse angle between the lines x-y+2=0 and (2-sqrt(3))x-y+1=0 .
Solve (sqrt(5)-1)/(sinx)+(sqrt(10+2sqrt(5)))/(cosx)=8,x in (0,pi/2)
Find the image of the circle x^2+y^2-2x+4y-4=0 in the line 2x-3y+5=0
The points on the line x=2 from which the tangents drawn to the circle x^2+y^2=16 are at right angles is (are) (a) (2,2sqrt(7)) (b) (2,2sqrt(5)) (c) (2,-2sqrt(7)) (d) (2,-2sqrt(5))
Calcualte the angle between the straight lines x+sqrt3 y+2=0 and y+5=0
The equation of a circle is x^2+y^2=4. Find the center of the smallest circle touching the circle and the line x+y=5sqrt(2)
A square is inscribed in the circle x^2+y^2-2x+4y+3=0 . Its sides are parallel to the coordinate axes. One vertex of the square is (a) (1+sqrt(2),-2) (b) (1-sqrt(2),-2) (c) (1,-2+sqrt(2)) (d) none of these
Find the length of the chord intercepted by the circle x^2+y^2-6x+8y-5=0 on the line 2x-y=5.
sinx+cosx=y^2-y+a has no value of x for any value of y if a belongs to (a) (0,""sqrt(3)) (b) (-""sqrt(3),0) (c) (-oo,-""sqrt(3)) (d) (""sqrt(3),oo)
If the circles x^2+y^2+2a x+c y+a=0 and x^2+y^2-3a x+d y-1=0 intersects at points P and Q , then find the values of a for which the line 5x+b y-a=0 passes through Pa n dQdot
