Question

All of the following is true except, if:
(i) $AC.AD=A{B}^2$ and
(ii) $AC.AD=B{C}^2$

• A. The points of intersection of direct common tangents and indirect common tangents of two circles divide the line segment joining the two centres respectively externally and internally in the ratio of their radii
  
• B. In a cyclic quadrilateral ABCD, if the diagonal CA bisects the angle C, then diagonal BD is parallel to the tangent at A to the circle through A, B, C, D
• C. If TA, TB are tangent segments to a circle C(O, r) from an external point T and OT intersects the circle in P, then AP bisects the angle TAB.
• D. If in a right triangle ABC, BD is the perpendicular on the hypotenuse AC, then

Answer 1

The correct option is D If in a right triangle ABC, BD is the perpendicular on the hypotenuse AC, then



$AC.AD=A{B}^{2}AC.AD=B{C}^2\Delta ABC\sim \Delta ADB\therefore \frac{AC}{AB}=\frac{AB}{AD}$
(Corresponding sides of similar triangle are proportional)
$AC.AD=A{B}^2\dots \left(1\right)$
Also, $\Delta ABC\sim \Delta BDC\Rightarrow \frac{AC}{BC}=\frac{BC}{CD}AC.CD=B{C}^2\dots \left(2\right)$
Both conditions of option (d) are found, therefore (d) is the answer.