SATHEE: Circle 4 Question 23

Circle 4 Question 23

23. Find the equation of the circle which passes through the point $(2, 0)$ and whose centre is the limit of the point of intersection of the lines $3 x + 5 y = 1, (2 + c) x + 5 c^{2} y = 1$ as $c$ tends to 1.

$(1979, 3 M)$

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Answer:

Correct Answer: 23. $25 (x^{2} + y^{2}) - 20 x + 2 y - 60 = 0$

Solution:

Given lines are

$3 x + 5 y - 1 = 0$

and $(2 + c) x + 5 c^{2} y - 1 = 0$

$∴ \frac{x}{- 5 + 5 c^{2}} = \frac{y}{- (2 + c) + 3} = \frac{1}{15 c^{2} - 5 c - 10}$

$\Rightarrow x = \frac{5 (c^{2} - 1)}{5 (3 c^{2} - c - 2)}$ and $y = \frac{1 - c}{15 c^{2} - 5 c - 10}$

$\Rightarrow lim_{c \to 1} x = lim_{c \to 1} \frac{2 c}{6 c - 1}$

and

$lim_{c \to 1} y = lim_{c \to 1} \frac{- 1}{30 c - 5} \Rightarrow lim_{c \to 1} x = \frac{2}{5}$

and

$lim_{c \to 1} y = - \frac{1}{25}$

$∴$ Centre $= lim_{c \to 1} x, lim_{c \to 1} y = \frac{2}{5}, - \frac{1}{25}$

$∴$ Radius $= \sqrt{2 - {\frac{2}{5}}^{2} + 0 + {\frac{1}{25}}^{2}} = \sqrt{\frac{64}{25} + \frac{1}{625}} = \frac{\sqrt{1601}}{25}$ $∴$ Equation of the circle is $x - {\frac{2}{5}}^{2} + y + {\frac{1}{25}}^{2} = \frac{1601}{625}$

$\begin{aligned} \Rightarrow & x^{2} + y^{2} - \frac{4 x}{5} + \frac{2 y}{25} + \frac{4}{25} + \frac{1}{625} - \frac{1601}{625} & = 0 \\ \Rightarrow & 25 (x^{2} + y^{2}) - 20 x + 2 y - 60 & = 0 \end{aligned}$

Circle 4 Question 23

23. Find the equation of the circle which passes through the point $(2, 0)$ and whose centre is the limit of the point of intersection of the lines $3 x + 5 y = 1, (2 + c) x + 5 c^{2} y = 1$ as $c$ tends to 1.

Answer:

Solution:

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Circle 4 Question 23

23. Find the equation of the circle which passes through the point (2,0) and whose centre is the limit of the point of intersection of the lines 3x+5y=1,(2+c)x+5c2y=1 as c tends to 1.

Answer:

Solution:

Engineering Preparation

Medical Preparation

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Important Resources

Other Resources

Syllabus

Test Platform

Sample Mock Test

Mentorship

NCERT Exercise Video Solution

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23. Find the equation of the circle which passes through the point $(2, 0)$ and whose centre is the limit of the point of intersection of the lines $3 x + 5 y = 1, (2 + c) x + 5 c^{2} y = 1$ as $c$ tends to 1.