SATHEE: Circle 1 Question 3

Circle 1 Question 3

3. If a circle of radius $R$ passes through the origin $O$ and intersects the coordinate axes at $A$ and $B$ , then the locus of the foot of perpendicular from $O$ on $A B$ is

(a) ${(x^{2} + y^{2})}^{2} = 4 R^{2} x^{2} y^{2}$

(b) ${(x^{2} + y^{2})}^{3} = 4 R^{2} x^{2} y^{2}$

(d) ${(x^{2} + y^{2})}^{2} = 4 R x^{2} y^{2}$

(2019 Main, 12 Jan II)

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Answer:

Correct Answer: 3. (b)

Solution:

Let the foot of perpendicular be $P (h, k)$ . Then, the slope of line $O P = \frac{k}{h}$

$∵$ Line $A B$ is perpendicular to line $O P$ , so slope of line

$A B = - \frac{h}{k}$

$[∵$ product of slopes of two perpendicular lines is $(- 1)$ ]

Now, the equation of line $A B$ is

$\begin{aligned} y - k = - \frac{h}{k} (x - h) \Rightarrow h x + k y = h^{2} + k^{2} \\ \frac{x}{\frac{h^{2} + k^{2}}{h}} + \frac{y}{\frac{h^{2} + k^{2}}{k}} = 1 \end{aligned}$

So, point $A \frac{h^{2} + k^{2}}{h}, 0$ and $B 0, \frac{h^{2} + k^{2}}{k}$

$∵ △ A O B$ is a right angled triangle, so $A B$ is one of the diameter of the circle having radius $R$ (given).

$\Rightarrow A B = 2 R$

$\begin{aligned} \Rightarrow \sqrt{\frac{h^{2} + k^{2}}{h} + \frac{h^{2} + k^{2}}{k}} = 2 R \\ \Rightarrow {(h^{2} + k^{2})}^{2} \frac{1}{h^{2}} + \frac{1}{k^{2}} = 4 R^{2} \\ \Rightarrow {(h^{2} + k^{2})}^{3} = 4 R^{2} h^{2} k^{2} \end{aligned}$

On replacing $h$ by $x$ and $k$ by $y$ , we get

${(x^{2} + y^{2})}^{3} = 4 R^{2} x^{2} y^{2},$

which is the required locus.

Circle 1 Question 3

3. If a circle of radius $R$ passes through the origin $O$ and intersects the coordinate axes at $A$ and $B$ , then the locus of the foot of perpendicular from $O$ on $A B$ is

Answer:

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Circle 1 Question 3

3. If a circle of radius R passes through the origin O and intersects the coordinate axes at A and B, then the locus of the foot of perpendicular from O on AB is

Answer:

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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3. If a circle of radius $R$ passes through the origin $O$ and intersects the coordinate axes at $A$ and $B$ , then the locus of the foot of perpendicular from $O$ on $A B$ is