CAT 2020

Slot 2

QA

Question & Solution

We can form the following figure based on the given information:

Since OA = 4 m and OB=3 m; AB = 5 m. OR bisects the chord into PC and QC. 

Since AB = 5 m, we have $a+b = 5      ...(i)$  Also, $4^2\ -k^2=a^2...\left(ii\right)$ and $3^2\ -k^2=b^2...\left(iii\right)$

Subtracting (iii) from (ii), we get: $a^2\ -b^2=7...\left(iv\right)$

Substituting (i) in (iv), we get $a - b = 1.4      ...(v)$; $\left[\left(a+b\right)\left(a\ -b\right)=7;\ \therefore\ \left(a-b\right)=\frac{7}{5}\right]$ 

Solving (i) and (v), we obtain the value of $a=3.2$ and $b=1.8$

Hence, $k^2\ =\ 5.76$

Moving on to the larger triangle $\triangle\ POC$, we have $5^2-k^2=\left(x+a\right)^2$; 

Substituting the previous values, we get: $(25-5.76)=\left(x+3.2\right)^2$ 

$\sqrt{19.24}=\left(x+3.2\right)$ or $x = 1.19 m$

Similarly, solving for y using $\triangle\ QOC$, we get $y=2.59 m$

Therefore, $PQ = 5+2.59+1.19 = 8.78 \approx\ 8.8 m$

Hence, Option A is the correct answer.