The length of the tangent from the point `(4, 5)` to the circle `x^2 + y^2 + 2x - 6y = 6` is : (A) `sqrt(13)` (B) `sqrt(38)` (C) `2sqrt(2)` (D) `2sqrt(13)`

If the length of the tangent from (2,5) to the circle x^(2) + y^(2) - 5x +4y + k = 0 is sqrt(37) then find k.

Length of the tangent. Prove that the length t o f the tangent from the point P (x_(1), y(1)) to the circle x^(2) div y^(2) div 2gx div 2fy div c = 0 is given by t=sqrt(x_(1)^(2)+y_(1)^(2)+2gx_(1)+2fy_(1)+c) Hence, find the length of the tangent (i) to the circle x^(2) + y^(2) -2x-3y-1 = 0 from the origin, (2,5) (ii) to the circle x^(2)+y^(2)-6x+18y+4=-0 from the origin (iii) to the circle 3x^(2) + 3y^(2)- 7x - 6y = 12 from the point (6, -7) (iv) to the circle x^(2) + y^(2) - 4 y - 5 = 0 from the point (4, 5).

The length of the tangent from a point on the circle x^(2)+y^(2)+4x-6y-12=0 to the circle x^(2)+y^(2)+4x-6y+4=0 is

The shortest distance between the parabola y^2 = 4x and the circle x^2 + y^2 + 6x - 12y + 20 = 0 is : (A) 0 (B) 1 (C) 4sqrt(2) -5 (D) 4sqrt(2) + 5

The image of the centre of the circle x^2 + y^2 = 2a^2 with respect to the line x + y = 1 is : (A) (sqrt(2), sqrt(2) (B) (1/sqrt(2) , sqrt(2) ) (C) (sqrt(2), 1/sqrt(2)) (D) none of these

The line 2x-y+1=0 is tangent to the circle at the point (2, 5) and the center of the circle lies on x-2y=4. The radius of the circle is (a) 3sqrt(5) (b) 5sqrt(3) (c) 2sqrt(5) (d) 5sqrt(2)

The radius of the circle touching the straight lines x-2y-1=0 and 3x-6y+7=0 is (A) 1/sqrt(2) (B) sqrt(5)/3 (C) sqrt(3) (D) sqrt(5)

The distance of the point P(1,-3) from the line 2y-3x=4 is (A) 13 (B) (7)/(13)sqrt(3) (C) sqrt(13) (D) (7)/(13)

The length of the chord of the parabola y^2=x which is bisected at the point (2, 1) is (a) 2sqrt(3) (b) 4sqrt(3) (c) 3sqrt(2) (d) 2sqrt(5)

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))