Question

ABC is an isosceles triangle. A circle is such that it passes through vertex C and AB acts as a tangent at D for the same circle. AC and BC intersect the circle at E and F respectively. AC = BC = 4 cm and AB = 6 cm. Also D is the mid-point of AB. What is the ratio of EC : (AE + AD)?

 

(a) 1 : 2 

(b) 1 : 3 

(c) 2 : 5 

(d)None of these

Answer 1

Answer : (b) 1 : 3 

Here AC and BC are the secants of the circle and AB is the tangent at D.

∴ AE × AC = AD2 ⇒ AE × 4 = (3)2 

⇒ AE = 9/4

∴ CE = AC – AE = 4 – \(\frac{9}{4}\) = \(\frac{7}{4}\) 

∴ CE : (AE + AD) = \(\frac{7}{4}\) : \(\big(\)\(\frac{9}{4}\)+3\(\big)\) 

= \(\frac{7}{4}\) : \(\frac{21}{4}\) = 1 : 3