Question
If (a + b + c + d) (a – b – c + d) = (a + b – c – d) (a – b + c – d), prove that a: b = c: d.
✓
Answer
Given $\frac{a+b+c+d}{a-b-c+d}=\frac{a+b-c-d}{a-b+c-d}$
Applying componendo and dividendo
$\frac{(a+b+c+d)+(a+b-c-d)}{(a+b+c+d)-(a+b-c-d)}=\frac{(a-b+c-d)+(a-b-c+d)}{(a-b+c-d)-(a-b-c+d)} $
$ \frac{2(a+b)}{2(c+d)}=\frac{2(a-b)}{2(c-d)} $
$\frac{a+b}{c+d}=\frac{a-b}{c-d} $
$ \frac{a+b}{a-b}=\frac{c+d}{c-d}$
Applying componendo and dividendo
$\frac{a+b+a-b}{a+b-a-b}=\frac{c+d+c-d}{c+d-c+d}$
$\frac{2 a}{2 b}=\frac{2 c}{2 d}$
$\frac{a}{b}=\frac{c}{d}$
Applying componendo and dividendo
$\frac{(a+b+c+d)+(a+b-c-d)}{(a+b+c+d)-(a+b-c-d)}=\frac{(a-b+c-d)+(a-b-c+d)}{(a-b+c-d)-(a-b-c+d)} $
$ \frac{2(a+b)}{2(c+d)}=\frac{2(a-b)}{2(c-d)} $
$\frac{a+b}{c+d}=\frac{a-b}{c-d} $
$ \frac{a+b}{a-b}=\frac{c+d}{c-d}$
Applying componendo and dividendo
$\frac{a+b+a-b}{a+b-a-b}=\frac{c+d+c-d}{c+d-c+d}$
$\frac{2 a}{2 b}=\frac{2 c}{2 d}$
$\frac{a}{b}=\frac{c}{d}$
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
In the given figure, $AB$ and $DE$ are perpendiculars to $BC.$
Find the ratio of the area of a $\triangle ABC :$ area of $\triangle DEC.$
→Find the ratio of the area of a $\triangle ABC :$ area of $\triangle DEC.$
Given $A=\left[\begin{array}{cc}1 & 1 \\ -2 & 0\end{array}\right]$ and $B=\left[\begin{array}{cc}2 & -1 \\ 1 & 1\end{array}\right] 3X + B + 2A = O$
→The percentage of marks obtained by a student in monthly unit tests are given below:
Based on this data find the probability that the student gets
(iii) less than 65% marks in a unit test.
| Unit Test | l | ll | lll | lV | V | Vl |
| Percentage of marks obtained | 72 | 67 | 69 | 74 | 71 | 76 |
(iii) less than 65% marks in a unit test.
Solve the following inequation, write the solution set and represent it on the number line.
$-3(x-7) \geq 15-7 x>\frac{x+1}{3}, x \in R$
→$-3(x-7) \geq 15-7 x>\frac{x+1}{3}, x \in R$
A page from the Savings Bank Account of Mr. Prateek is given below :
If he receives ₹198 as interest on $1^{st}$ January $2007$, find the rate of interest paid by the bank.
→| Date | Particulars | Withdrawals (in ₹) |
Deposits (in ₹) |
Balance (in ₹) |
| January $1^{st}, 2006$ | B/F | - | - | $1,270$ |
| January $7^{th}, 2006$ | By Cheque | - | $2,310$ | $3,580$ |
| March $9^{th}, 2006$ | To Self | $2,000$ | - | $1,580$ |
| March $26^{th}, 2006$ | By Cash | - | $6,200$ | $7,780$ |
| June $10^{th}, 2006$ | To Cheque | $4,500$ | - | $3,280$ |
| July $15^{th}, 2006$ | By Clearing | - | $2,630$ | $5,910$ |
| October $18^{th}, 2006$ | To Cheque | $530$ | - | $5,380$ |
| October $27^{th}, 2006$ | To Self | $2,690$ | - | $2,690$ |
| November $3^{rd}, 2006$ | By Cash | - | $1,500$ | $4,190$ |
| December $6^{th}, 2006$ | To Cheque | $950$ | $3,240$ | |
| December $23^{rd}, 2006$ | By Transfer | - | $2,920$ | $6,160$ |
If he receives ₹198 as interest on $1^{st}$ January $2007$, find the rate of interest paid by the bank.
All the three face cards of spades are removed from a well shuffled pack of 52 cards. A card is then drawn at random from the remaining pack. Find the probability of getting: a queen
Which term of A.P. $5, 15, 25 …………$ will be $130$ more than its $31st$ term?
→$
\text { prove that } \frac{1}{1+\cos \left(90^{\circ}-A\right)}+\frac{1}{1-\cos \left(90^{\circ}-A\right)}=2 \operatorname{cosec} c^2\left(90^{\circ}-A\right)
$
→\text { prove that } \frac{1}{1+\cos \left(90^{\circ}-A\right)}+\frac{1}{1-\cos \left(90^{\circ}-A\right)}=2 \operatorname{cosec} c^2\left(90^{\circ}-A\right)
$
The difference between the squares of two numbers is 45. The square of the smaller number is 4 times the larger number. Determine the numbers.
Find the mode of the following frequency distribution:
| x | 10 | 11 | 12 | 13 | 14 | 15 |
| f | 1 | 4 | 7 | 5 | 9 | 3 |