Question
Two circles with centres O and O' intersect at two points A and B. A line PQ is drawn parallel to OO' through A or B, intersecting the circles at P and Q. Prove that PQ = 2OO'.
✓
Answer
Given: Two circles with centres O and O' intersect at two points A and B. 
Draw a line PQ parallel to OO' through B, OX perpendicular to PQ, O'Y perpendicular to PQ, join all. We know that perpendicular drawn from the centre to the chord, bisects the chord.$\therefore$ PX = XB and YQ = BY
$\therefore$ PX + YQ = XB + BY
On adding XB + BY on both sides, we get PX + YQ + XB + BY = 2(XB + BY) ⇒ PQ = 2(XY) ⇒ PQ = 2(OO') Hence, PQ = 2OO'
Draw a line PQ parallel to OO' through B, OX perpendicular to PQ, O'Y perpendicular to PQ, join all. We know that perpendicular drawn from the centre to the chord, bisects the chord.$\therefore$ PX = XB and YQ = BY$\therefore$ PX + YQ = XB + BY
On adding XB + BY on both sides, we get PX + YQ + XB + BY = 2(XB + BY) ⇒ PQ = 2(XY) ⇒ PQ = 2(OO') Hence, PQ = 2OO'
Need a full question paper?
Generate a complete, print-ready paper with questions like this in minutes — across 16+ boards, with answer keys.
Explore more
Similar questions
Simplify the following:$\Big(\text{x}+\frac{2}{\text{x}}\Big)^3+\Big(\text{x}-\frac{2}{\text{x}}\Big)^3$
→In figure, find the area of $\triangle\text{GEF}.$
→In □ABCD, side BC < side AD, side BC || side AD and if side BA ≅ side CD, then prove that ∠ABC = ∠DCB.
Given: side BC < side AD, side BC || side AD, side BA = side CD
To prove: ∠ABC ≅ ∠DCB
Construction: Draw seg BP ⊥ side AD, A – P – D
seg CQ ⊥ side AD, A – Q – D
Use the given letters to write the answer.
i. There are ‘a’ trees in the village Lat. If the number of trees increases every year by ’b‘. then how many trees will there be after ‘x’ years?
ii. For the parade there are y students in each row and x such row are formed. Then, how many students are there for the parade in all ?
iii. The tens and units place of a two digit number is m and n respectively. Write the polynomial which represents the two digit number.
i. There are ‘a’ trees in the village Lat. If the number of trees increases every year by ’b‘. then how many trees will there be after ‘x’ years?
ii. For the parade there are y students in each row and x such row are formed. Then, how many students are there for the parade in all ?
iii. The tens and units place of a two digit number is m and n respectively. Write the polynomial which represents the two digit number.
Divide each of the following polynomials by synthetic division method and also by linear division method. Write the quotient and the remainder : $(2x^4 + 3x^3 + 4x – 2x^2) ÷ (x + 3)$
→If $2$ and $0$ are the zeros of the polynomial $f(x) = 2x^3 - 5x^2 + ax + b$ then find the values of a and b.
Hint: $f(x) = 0$ and $f(0) = 0.$
→Hint: $f(x) = 0$ and $f(0) = 0.$
In the given figure, ABCD is a square and P is a point inside it such that PB = PD. Prove that CPA is a straight line.
In the adjoining figure, ABCD is a quadrilateral and AC is one of its diagonals. Prove that:
(i) AB + BC + CD + DA > 2AC
(ii) AB + BC + CD > DA
(iii) AB + BC + CD + DA > AC + BD
(i) AB + BC + CD + DA > 2AC
(ii) AB + BC + CD > DA
(iii) AB + BC + CD + DA > AC + BD
For the above given activity, the information of number of girls per 1000 boys is given for five states. The literacy percentage of these five states is given below. Assam (73%), Bihar (64%), Punjab (77%), Kerala (94%), Maharashtra (83%). Think of the number of girls and the literacy percentages in the respective states. Can you draw any conclusions from it?
Given below is the bar graph indicating the marks obtained out of 50 in mathematics paper by 100 students. Read the bar graph and answer the following questions:
- It is decided to distribute workbooks on mathematics to the students obtaining less than 20 marks, giving one workbook to each of such students. If a workbook costs Rs. 5, what sum is required to buy the workbooks?
- Every student belonging to the highest mark group is entitled to get a prize of Rs. 10. How much amount of money is required for distributing the prize money?
- Every student belonging to the lowest mark-group has to solve 5 problems per day. How many problems, in all, will be solved by the students of this group per day?
- State whether true or false:
- 17% students have obtained marks ranging from 40 to 49.
- 59 students have obtained marks ranging from 10 to 29.
- What is the number of students getting less than 20 marks?
- What is the number of students getting more than 29 marks?
- What is the number of students getting marks between 9 and 40?
- What is the number of students belonging to the highest mark group?
- What is the number of students obtaining more than 19 marks?