Applyo - College Application Platform

CAT 2020 Slot 2 QA Question & Solution

GeometryMedium

Question

Let C be a circle of radius 5 meters having center at O. Let PQ be a chord of C that passes through points A and B where A is located 4 meters north of O and B is located 3 meters east of O. Then, the length of PQ, in meters, is nearest to

Options

8.8
7.8
6.6
7.2

Solution

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

Question Figure

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.