Statement 1 :The circles `x^2+y^2+2p x+r=0` and `x^2+y^2+2q y+r=0` touch if `1/(p^2)+1/(q^2)=1/r dot` Statement 2 : Two centers `C_1a n dC_2` and radii `r_1a n dr_2,` respectively, touch each other if `|r_1+-r_2|=c_1c_2dot`

Let A B C be a triangle right-angled at Aa n dS be its circumcircle. Let S_1 be the circle touching the lines A B and A C and the circle S internally. Further, let S_2 be the circle touching the lines A B and A C produced and the circle S externally. If r_1 and r_2 are the radii of the circles S_1 and S_2 , respectively, show that r_1r_2=4 area ( A B C)dot

If the radii of the circles (x-1)^2+(y-2)^2=1 and (x-7)^2+(y-10)^2=4 are increasing uniformly w.r.t. time as 0.3 units/s and 0.4 unit/s, respectively, then at what value of t will they touch each other?

Let C_1 be the circle with center O_1(0,0) and radius 1 and C_2 be the circle with center O_2(t ,t^2+1),(t in R), and radius 2. Statement 1 : Circles C_1a n dC_2 always have at least one common tangent for any value of t Statement 2 : For the two circles O_1O_2geq|r_1-r_2|, where r_1a n dr_2 are their radii for any value of tdot

C is the center of the hyperbola (x^2)/(a^2)-(y^2)/(b^2)=1 The tangent at any point P on this hyperbola meet the straight lines b x-a y=0 and b x+a y=0 at points Qa n dR , respectively. Then prove that C QdotC R=a^2+b^2dot

If the two circles (x+1)^2+(y-3)=r^2 and x^2+y^2-8x+2y+8=0 intersect in two distinct point,then (A) r > 2 (B) 2 < r < 8 (C) r < 2 (D) r=2

Statement-l: If the diagonals of the quadrilateral formed by the lines px+ gy+ r= 0, p'x +gy +r'= 0, p'x +q'y +r'= 0 , are at right angles, then p^2 +q^2= p'^2+q'^2 . Statement-2: Diagonals of a rhombus are bisected and perpendicular to each other.

Let C_1 and C_2 be parabolas x^2 = y - 1 and y^2 = x-1 respectively. Let P be any point on C_1 and Q be any point C_2 . Let P_1 and Q_1 be the reflection of P and Q, respectively w.r.t the line y = x then prove that P_1 lies on C_2 and Q_1 lies on C_1 and PQ >= [P P_1, Q Q_1] . Hence or otherwise , determine points P_0 and Q_0 on the parabolas C_1 and C_2 respectively such that P_0 Q_0 <= PQ for all pairs of points (P,Q) with P on C_1 and Q on C_2

Statement 1: If b c+q r=c a+r p=a b+p q=-1, t h e n|a p a p b q b q c r c r|=0(a b c ,p q r!=0)dot Statement 2: if system of equations a_1x+b_1y+c_1=0,a_2x+b_2y+c_2=0,a_3x+b_2y+c^3=0 has non -trivial solutions |a_1b_1c_1a_2b_2c_2a_3b_3c_3|=0

The circle C_(1):x^(2)+y^(2)=3 , with centre at O, intersects the parabola x^(2)=2y at the point P in the quadrant. Let the tangent to the circle C_(1) at P touches other tqo circles C_(2)andC_(3) at R_(2)andR_(3) , respectively. Suppose C_(2)andC_(3) have equal radii 2sqrt(3) and centres Q_(2)andQ_(3) . respectively. If Q_(2)andQ_(3) lie on the y-axis, then

Statement 1: ^m C_r+^m C_(r-1)^n C_1+^mC_(r-2)^n C_2++^n C_r=0,ifm+n