SATHEE: Circle 5 Question 1

Circle 5 Question 1

1. If the angle of intersection at a point where the two circles with radii $5 c m$ and $12 c m$ intersect is $90^{\circ}$ , then the length (in cm) of their common chord is

(a) $\frac{13}{5}$

(b) $\frac{120}{13}$

(d) $\frac{13}{2}$

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Answer:

Correct Answer: 1. (b)

Solution:

Let the length of common chord $= A B = 2 A M = 2 x$

Now, $C_{1} C_{2} = \sqrt{A C_{1}^{2} + A C_{2}^{2}}$

$[∵$ circles intersect each other at $90^{\circ}]$

$\begin{aligned} and C_{1} C_{2} = C_{1} M + M C_{2} \\ \Rightarrow C_{1} C_{2} = \sqrt{12^{2} - A M^{2}} + \sqrt{5^{2} - A M^{2}} \end{aligned}$

From Eqs. (i) and (ii), we get

$\begin{aligned} \sqrt{A C_{1}^{2} + A C_{2}^{2}} = \sqrt{144 - A M^{2}} + \sqrt{25 - A M^{2}} \\ \Rightarrow \sqrt{144 + 25} = \sqrt{144 - x^{2}} + \sqrt{25 - x^{2}} \\ \Rightarrow 13 = \sqrt{144 - x^{2}} + \sqrt{25 - x^{2}} \end{aligned}$

On squaring both sides, we get

$\begin{aligned} 169 = 144 - x^{2} + 25 - x^{2} + 2 \sqrt{144 - x^{2}} \sqrt{25 - x^{2}} \\ \Rightarrow x^{2} = \sqrt{144 - x^{2}} \sqrt{25 - x^{2}} \end{aligned}$

Again, on squaring both sides, we get

$x^{4} = (144 - x^{2}) (25 - x^{2}) = (144 \times 25) - (25 + 144) x^{2} + x^{4}$

$\Rightarrow x^{2} = \frac{144 \times 25}{169} \Rightarrow x = \frac{12 \times 5}{13} = \frac{60}{13} c m$

Now, length of common chord $2 x = \frac{120}{13} c m$

Alternate Solution

Given, $A C_{1} = 12 c m$ and $A C_{2} = 5 c m$

In $Δ C_{1} A C_{2}$ ,

$C_{1} C_{2} = \sqrt{{(C_{1} A)}^{2} + {(A C_{2})}^{2}} [∵ ∠ C_{1} A C_{2} = 90^{\circ},$

because circles intersects each other at $90^{\circ}$ ]

$= \sqrt{169} = 13 c m$

$= \sqrt{(12)^{2} + (5)^{2}} = \sqrt{144 + 25}$

Now, area of $Δ C_{1} A C_{2} = \frac{1}{2} A C_{1} \times A C_{2}$

$= \frac{1}{2} \times 12 \times 5 = 30 c m^{2}$

Also, area of $Δ C_{1} A C_{2} = \frac{1}{2} C_{1} C_{2} \times A M$

$= \frac{1}{2} \times 13 \times \frac{A B}{2} ∵ A M = \frac{A B}{2}$

$\begin{aligned} ∴ \frac{1}{4} \times 13 \times A M & = 30 c m \\ A M & = \frac{120}{13} c m \end{aligned}$

Circle 5 Question 1

1. If the angle of intersection at a point where the two circles with radii $5 c m$ and $12 c m$ intersect is $90^{\circ}$ , then the length (in cm) of their common chord is

Answer:

Alternate Solution

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Circle 5 Question 1

1. If the angle of intersection at a point where the two circles with radii 5cm and 12cm intersect is 90∘, then the length (in cm) of their common chord is

Answer:

Alternate Solution

Engineering Preparation

Medical Preparation

NCERT Books & Solutions

Important Resources

Other Resources

Syllabus

Test Platform

Sample Mock Test

Mentorship

NCERT Exercise Video Solution

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1. If the angle of intersection at a point where the two circles with radii $5 c m$ and $12 c m$ intersect is $90^{\circ}$ , then the length (in cm) of their common chord is