Paragraph: A rectangle P Q R S is inscribed in a circle C with centre O and a diameter along the line 2 y-4 x=9 . P and Q are the points (-8,-2) and (4,-2) respectively. Let a circle C₁ be drawn taken RS as the diameter. On the basis of above information, answer the following : Question: If m₁ & m₂ are the finite slopes of tangent to circle C₁ from P & Q respectively, then the value of | m ₁ |+ | m ₂ | is -

Step-by-step solution

Let O ( h , k ) be the centre of the circle C aligned & 2 k -4 ~h =9 . . . (i) & Also OP = OQ = Radius of the circle & OP ^2= OQ ^2 or h +8= ( h -4) & h +8=- h +4 ~h =-2 aligned Putting in (i), k = 1 2 array ll & Radius = OP = 13 2 & P R=13 array aligned & In PQR , PR ^2= PQ ^2+ QR ^2 & 169=144+ QR ^2 QR =5 aligned Also co-ordinates of R & ~S are respectively (4,3) &(-8,3) Now C₁:(x-4)(x+8)+(y-3)(y-3)=0 or x^2+y^2+4 x-6 y-23=0 centre (-2,3) & Radius =6 Equation of tangent through P to C ₁ is y+2=m₁(x+8) | 6 ~m ₁-5