SATHEE: Circle 2 Question 1

Circle 2 Question 1

1. If the circles $x^{2} + y^{2} + 5 K x + 2 y + K = 0$ and $2 (x^{2} + y^{2}) + 2 K x + 3 y - 1 = 0, (K \in R)$ , intersect at the points $P$ and $Q$ , then the line $4 x + 5 y - K = 0$ passes through $P$ and $Q$ , for

(2019 Main, 10 April I)

(a) no values of $K$

(b) exactly one value of $K$

(d) infinitely many values of $K$

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Answer:

Correct Answer: 1. (a)

Solution:

Equation of given circles

$\begin{array}{lc} x^{2} + y^{2} + 5 K x + 2 y + K = 0 \\ and & 2 (x^{2} + y^{2}) + 2 K x + 3 y - 1 = 0 \\ \Rightarrow & x^{2} + y^{2} + K x + \frac{3}{2} y - \frac{1}{2} = 0 \end{array}$

On subtracting Eq. (ii) from Eq. (i), we get

$\begin{aligned} 4 K x + \frac{1}{2} y + K + \frac{1}{2} & = 0 \\ \Rightarrow 8 K x + y + (2 K + 1) & = 0 \end{aligned}$

$[∵$ if $S_{1} = 0$ and $S_{2} = 0$ be two circles, then their common chord is given by $S_{1} - S_{2} = 0$ .]

Eq. (iii) represents equation of common chord as it is given that circles (i) and (ii) intersects each other at points $P$ and $Q$ .

Since, line $4 x + 5 y - K = 0$ passes through point $P$ and $Q$ .

$∴ \frac{8 K}{4} = \frac{1}{5} = \frac{2 K + 1}{- K}$

$\Rightarrow K = \frac{1}{10}$ [equating first and second terms]

and $- K = 10 K + 5$

[equating second and third terms]

$\Rightarrow 11 K + 5 = 0 \Rightarrow K = - \frac{5}{11}$

$∵ \frac{1}{10} \neq - \frac{5}{11}$ , so there is no such value of $K$ , for which line $4 x + 5 y - K = 0$ passes through points $P$ and $Q$ .

Circle 2 Question 1

1. If the circles $x^{2} + y^{2} + 5 K x + 2 y + K = 0$ and $2 (x^{2} + y^{2}) + 2 K x + 3 y - 1 = 0, (K \in R)$ , intersect at the points $P$ and $Q$ , then the line $4 x + 5 y - K = 0$ passes through $P$ and $Q$ , for

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Circle 2 Question 1

1. If the circles x2+y2+5Kx+2y+K=0 and 2(x2+y2)+2Kx+3y−1=0,(K∈R), intersect at the points P and Q, then the line 4x+5y−K=0 passes through P and Q, for

Answer:

Engineering Preparation

Medical Preparation

NCERT Books & Solutions

Important Resources

Other Resources

Syllabus

Test Platform

Sample Mock Test

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NCERT Exercise Video Solution

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1. If the circles $x^{2} + y^{2} + 5 K x + 2 y + K = 0$ and $2 (x^{2} + y^{2}) + 2 K x + 3 y - 1 = 0, (K \in R)$ , intersect at the points $P$ and $Q$ , then the line $4 x + 5 y - K = 0$ passes through $P$ and $Q$ , for