SATHEE: Circle 4 Question 20

Circle 4 Question 20

20. Let $S \equiv x^{2} + y^{2} + 2 g x + 2 f y + c = 0$ be a given circle. Find the locus of the foot of the perpendicular drawn from the origin upon any chord of $S$ which subtends a right angle at the origin.

$(1988, 5$ M)

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Answer:

Correct Answer: 20. $x^{2} + y^{2} + g x + f y + \frac{c}{2} = 0$

Solution:

Let $P (h, k)$ be the foot of perpendicular drawn from origin $O (0, 0)$ on the chord $A B$ of the given circle such that the chord $A B$ subtends a right angle at the origin.

The equation of chord $A B$ is

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

The combined equation of $O A$ and $O B$ is homogeneous equation of second degree obtained by the help of the given circle and the chord $A B$ and is given by,

$x^{2} + y^{2} + (2 g x + 2 f y) \frac{h x + k y}{h^{2} + k^{2}} + c {\frac{h x + k y}{h^{2} + k^{2}}}^{2} = 0$

Since, the lines $O A$ and $O B$ are at right angle. $∴$ Coefficient of $x^{2} +$ Coefficient of $y^{2} = 0$

$\Rightarrow 1 + \frac{2 g h}{h^{2} + k^{2}} + \frac{c h^{2}}{{(h^{2} + k^{2})}^{2}}$

$+ 1 + \frac{2 f k}{h^{2} + k^{2}} + \frac{c k^{2}}{{(h^{2} + k^{2})}^{2}} = 0$

$\Rightarrow 2 (h^{2} + k^{2}) + 2 (g h + f k) + c = 0$

$\Rightarrow h^{2} + k^{2} + g h + f k + \frac{c}{2} = 0$

$∴$ Required equation of locus is

$x^{2} + y^{2} + g x + f y + \frac{c}{2} = 0$

Circle 4 Question 20

20. Let $S \equiv x^{2} + y^{2} + 2 g x + 2 f y + c = 0$ be a given circle. Find the locus of the foot of the perpendicular drawn from the origin upon any chord of $S$ which subtends a right angle at the origin.

Answer:

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Circle 4 Question 20

20. Let S≡x2+y2+2gx+2fy+c=0 be a given circle. Find the locus of the foot of the perpendicular drawn from the origin upon any chord of S which subtends a right angle at the origin.

Answer:

Engineering Preparation

Medical Preparation

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Important Resources

Other Resources

Syllabus

Test Platform

Sample Mock Test

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NCERT Exercise Video Solution

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20. Let $S \equiv x^{2} + y^{2} + 2 g x + 2 f y + c = 0$ be a given circle. Find the locus of the foot of the perpendicular drawn from the origin upon any chord of $S$ which subtends a right angle at the origin.