If a c =c d =c f prove that : (b^2+d^2+f^2 ) (a^2+c^2+e^2 )=(a b+… - Vidyadip

Question

If $\frac{a}{c}=\frac{c}{d}=\frac{c}{f}$ prove that $:\left(b^2+d^2+f^2\right)\left(a^2+c^2+e^2\right)=(a b+$ $c d+e f)^2$

✓

Answer

$\begin{aligned} & \frac{a}{c}=\frac{c}{d}=\frac{c}{f}= k \text { (say) } \\ & \therefore a=b k_t c=d k^2 e=f k \\ & \text { L.H.S. }=\left(b^2+d^2+f^2\right)\left(a^2+c^2+e^2\right) \\ & =\left(b^2+d^2+f^2\right)\left(b^2 k^2+d^2 k^2+f^2 k^2\right) \\ & =k^2\left(b^2+d^2+f^2\right) k^2\left(b^2+d^2+f^2\right) \\ & =k^2\left(b^2+d^2+f^2\right)^2 \\ & \text { R.H.S. }=(a b+c d+e f)^2 \\ & =(b . k b+d k \cdot d+f k . f)^2 \\ & =\left(k b^2+k d^2+k f^2\right) \\ & =k^2\left(b^2+d^2+f^2\right)^2 \\ & \therefore \text { L.H.S. }=\text { R.H.S. }\end{aligned}$

Need a full question paper?

Generate a complete, print-ready paper with questions like this in minutes — across 16+ boards, with answer keys.

Start Generating Free

Explore more

More "[4 marks sum]" from Ratio and Proportion→Ratio and Proportion — all question types→Mathematics STD 10 question papers→ICSE Board STD 10 question papers→ICSE Board question paper generator→

Similar questions

In what ratio does the point $(1, a)$ divided the join of $(-1, 4)$ and $( 4, -1)?$ Also, find the value of a.

A man has some shares of Rs. 100 par value paying 6% dividend. He sells half of these at a discount of 10% and invests the proceeds in 7% Rs. 50 shares at a premium of Rs. 10. This transaction decreases his income from dividends by Rs. 120. Calculate:
(i) the number of shares before the transaction.
(ii) the number of shares he sold.
(iii) his initial annual income from shares.

Bhavana invested Rs 20,000 and Rs 25,000 in buying shares of 'Bharati Telecom' and 'Satyam Infoways' which later declared dividend of 10% and 12.5% respectively. After collecting the dividends Bhavana sells all her shares at a loss of 4% and 5% respectively on her investments. Find her total earnings.

(i) write down the co-ordinates of the point $P$ that divides the line joining $A(-4, 1)$ and $B(17, 10)$ in the ratio $1 : 2.$
(ii) Calculate the distance $OP,$ where $O$ is the origin.
(iii) In what ratio does the $Y-$ axis divide the line $AB?$

Find the diameter of the circle if the length of a chord is $3.2 cm$ and itd distance from the centre is $1.2 cm$.

Let $A=\left[\begin{array}{cc}2 & 1 \\ 0 & -2\end{array}\right], B=\left[\begin{array}{cc}4 & 1 \\ -3 & -2\end{array}\right]$ and $C=\left[\begin{array}{ll}-3 & 2 \\ -1 & 4\end{array}\right]$. Find $A^2+A C-5 B$.

A buoy is made in the form of a hemisphere surmounted by a right circular cone whose circular base coincides with the plane surface of the hemisphere. The radius of the base of the cone is 3.5 m and its volume is two-third the volume of hemisphere. Calculate the height of the cone and the surface area of the buoy, correct to two decimal places.

The age of a father is twice the square of the age of his son. Eight years hence, the age of the father will be $4$ years more than three times the age of the son. Find their present ages.

If $5$th and $6$th terms of an A.P are respectively $6$ and $5.$ Find the 11th term of the A.P

An increase of 4% on the sales tax of an article led to an increase of Rs 480 on the selling price of the article. Find the marked price of the article.