SATHEE: Circle 5 Question 4

Circle 5 Question 4

4. The locus of the mid-point of the chord of contact of tangents drawn from points lying on the straight line $4 x - 5 y = 20$ to the circle $x^{2} + y^{2} = 9$ is

(2012)

(a) $20 (x^{2} + y^{2}) - 36 x + 45 y = 0$

(b) $20 (x^{2} + y^{2}) + 36 x - 45 y = 0$

(d) $36 (x^{2} + y^{2}) + 20 x - 45 y = 0$

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Answer:

Correct Answer: 4. (a)

Solution:

PLAN If $S : a x^{2} + 2 h x y + b y^{2} + 2 g x + 2 f y + C$ then equation of chord bisected at $P (x_{1}, y_{1})$ is $T = S_{1}$ or a $x x_{1} + h (x y_{1} + y x_{1}) + b y y_{1} + g (x + x_{1}) + f (y + y_{1}) + C$ $= a x_{1}^{2} + 2 h x_{1} y_{1} + b y_{1}^{2} + 2 g x_{1} + 2 f y_{1} + C$

Description of Situation As equation of chord of contact is $T = 0$

Here, equation of chord of contact w.r.t. $P$ is

and equation of chord bisected at the point $Q (h, k)$ is

$\begin{aligned} x h + y k - 9 & = h^{2} + k^{2} - 9 \\ \Rightarrow & x h + k y & = h^{2} + k^{2} \end{aligned}$

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

$\begin{aligned} \frac{5 λ}{h} & = \frac{4 λ - 20}{k} = \frac{45}{h^{2} + k^{2}} \\ ∴ & λ & = \frac{20 h}{4 h - 5 k} and λ = \frac{9 h}{h^{2} + k^{2}} \\ \Rightarrow & \frac{20 h}{4 h - 5 k} & = \frac{9 h}{h^{2} + k^{2}} \\ or & 20 (h^{2} + k^{2}) & = 9 (4 h - 5 k) \\ or & 20 (x^{2} + y^{2}) & = 36 x - 45 y \end{aligned}$

Circle 5 Question 4

4. The locus of the mid-point of the chord of contact of tangents drawn from points lying on the straight line $4 x - 5 y = 20$ to the circle $x^{2} + y^{2} = 9$ is

Answer:

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Circle 5 Question 4

4. The locus of the mid-point of the chord of contact of tangents drawn from points lying on the straight line 4x−5y=20 to the circle x2+y2=9 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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4. The locus of the mid-point of the chord of contact of tangents drawn from points lying on the straight line $4 x - 5 y = 20$ to the circle $x^{2} + y^{2} = 9$ is