Two different non-parallel lines cut the circle `|z| = r` at points `a, b, c and d`, respectively. Prove that these lines meet at the point `z` given by `(a^-1+b^-1-c^-1-d^-1)/(a^-1b^-1-c^-1d^-1)`

Two different non-parallel lines meet the circle abs(z)=r . One of them at points a and b and the other which is tangent to the circle at c. Show that the point of intersection of two lines is (2c^(-1)-a^(-1)-b^(-1))/(c^(-2)-a^(-1)b^(-1)) .

Three points represented by the complex numbers a,b,c lie on a circle with centre 0 and rdius r. The tangent at C cuts the chord joining the points a,b and z. Show that z= (a^-1+b^-1-2c^-1)/(a^-1b^-1-c^-2)

If a^x=b ,\ b^y=c\ a n d\ c^z=a , prove that x y z=1

If a + c = b e and 1/b + 1/d = e/c , Prove that : a,b,c and d are in proportion.

A line meets the coordinate axes at A and B . A circle is circumscribed about the triangle O A Bdot If d_1a n dd_2 are distances of the tangents to the circle at the origin O from the points Aa n dB , respectively, then the diameter of the circle is (a) (2d_1+d_2)/2 (b) (d_1+2d_2)/2 (c) d_1+d_2 (d) (d_1d_2)/(d_1+d_2)

Line segments A C and B D are diameters of the circle of radius 1. If /_B D C=60^0 , the length of line segment A B is_________

Direction ratios of two lines are a ,b ,c a n d1//b c ,1//c a ,1//a bdot Then the lines are ______.

Direction ratios of two lines are a ,b ,ca n d1//b c ,1//c a ,1//a bdot Then the lines are ______.

Two circles C_1 and C_2 intersect at two distinct points P and Q in a line passing through P meets circles C_1 and C_2 at A and B , respectively. Let Y be the midpoint of A B and Q Y meets circles C_1 and C_2 at X a n d Z respectively. Then prove that Y is the midpoint of X Z

Show that the line represented by equation x=ay+b,z=cy+d in symmetric form is (x-b)/a=y/1=(z-d)/c