$AB$ is a diameter of a circle with centre $O$. A line drawn through the point $A$ intersects the circle at the point $C$ and the tangent through $B$ at the point $D$. Prove that $\left(a\right)B{D}^2=AD.DC$ . $\left(b\right)$ The area of the rectangle of formed by $AC$ and $AD$ for any straight line is always equal.Solution in Bengali

In the following figure , $\angle P=$

In $\Delta PQR$ and $\Delta DEF$, $\frac{PQ}{DE}=\frac{QR}{EF}=\frac{RP}{FD}$. Then which one of the following is correct?

A person deposite ₹ 100 in a bank and after 2 yars he got a total ₹121 as a return. Then the componund interest of the bank per annum is

Goutam took a loan of some money from a cooperative bank for opening a poultry farm at the rate of simple interest of $12\%$ per annum. Every month he has to repay $378Rs$ as interest. Determine the loan amount taken by him.

Two circles intersect each other at the points P and Q. Two straight lines through P and Q intersect one circle at the points A and C and the other circle at B and D. Prove the AC|| BD

In the figure, O is the centre of the circle . Seg AB is the diameter at the point C on the circle the tangent CD is drawn. Line BD is a tangent to the circle at point B. Show that seg OD $\mid \mid$ chord AC.

The centre of two circle are P and Q they intersect at the points A and B. The straight line parallel to the line segment PQ through the point A intersects the two circles at the point C and D Prove that $CD=2PQ$

Two circles intersect each other at the points P and Q. Two straight lines through P and Q intersect on circle at the points A and C and the other circle at B and D . Prove that AC||BD.

In two circles , one circle passes through the centre O of the other circle and they intersect each other at the points A and B . A straight line passing through A intersect the circle passing through O at the point P and the circle with centre at O at the point R . BY joining P , B and R , B prove that PR= PB.

The straight line PAB intersects the circle with centre O at the points A and B. The straight line PAB will be a tangent to that circle if $AB=\text{\_\_\_\_\_\_\_}$

In the adjoining figure, O is the centre and seg AB is a diameter. At the point C on the circle, the tangent CD is drawn. Line BD is a tangent to the circle at the point B. Show that segOD $\mid \mid$ chord AC.

$AB$ is a diameter of a circle. $BP$ is a tangent to the circle at $B$. A straight line passing through $A$ intersects $BP$ at $C$ and the circle at $D$. Prove that $B{C}^2=AC\times CD$.

$AB$ is a diameter of a circle with centre $O$, $P$ is any point on the circle, the tangent drawn through the point $P$ intersects the two tangents drawn through the points $A$ and $B$ at the points $Q$ and $R$ respectively. If the radius of the circle be $r$, then prove that $PQ.PR=r^2$.

O কেন্দ্রীয় বৃত্তের AB ব্যাস। A বিন্দু দিয়ে অঙ্কিত একটি সরলরেখা বৃত্তকে C বিন্দুতে এবং B বিন্দুতে অঙ্কিত স্পর্শককে D বিন্দুতে ছেদ করে। প্রমাণ কর যে, BD^2=AD.DC

O কেন্দ্রীয় বৃত্তের AB ব্যাস। A বিন্দু দিয়ে অঙ্কিত একটি সরলরেখা বৃত্তকে C বিন্দুতে এবং B বিন্দুতে অঙ্কিত স্পর্শককে D বিন্দুতে ছেদ করে। প্রমাণ কর যে, যে কোনো সরলরেখার জন্য AC এবং AD দ্বারা গঠিত আয়তক্ষেত্রের ক্ষেত্রফল সর্বদা সমান।