SATHEE: Circle 2 Question 2

Circle 2 Question 2

2. If a tangent to the circle $x^{2} + y^{2} = 1$ intersects the coordinate axes at distinct points $P$ and $Q$ , then the locus of the mid-point of $P Q$ is

(2019 Main, 9 April I)

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

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

(d) $x^{2} + y^{2} - 16 x^{2} y^{2} = 0$

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Answer:

Correct Answer: 2. (c)

Solution:

Equation of given circle is $x^{2} + y^{2} = 1$ , then equation of tangent at the point $(\cos θ, \sin θ)$ on the given circle is

$x \cos θ + y \sin θ = 1$

$[∵$ Equation of tangent at the point $P (\cos θ, \sin θ)$ to the circle $x^{2} + y^{2} = r^{2}$ is $x \cos θ + y \sin θ = r]$

Now, the point of intersection with coordinate axes are $P (\sec θ, 0)$ and $Q (0, cosec θ)$ .

$∵$ Mid-point of line joining points $P$ and $Q$ is

$M \frac{\sec θ}{2}, \frac{cosec θ}{2} = (h, k) (let)$

So, $\cos θ = \frac{1}{2 h}$ and $\sin θ = \frac{1}{2 k}$

$∵ \sin^{2} θ + \cos^{2} θ = 1$

$∴ \frac{1}{4 h^{2}} + \frac{1}{4 k^{2}} = 1 \Rightarrow \frac{1}{h^{2}} + \frac{1}{k^{2}} = 4$

Now, locus of mid-point $M$ is

$\begin{aligned} \frac{1}{x^{2}} + \frac{1}{y^{2}} & = 4 \\ \Rightarrow x^{2} + y^{2} - 4 x^{2} y^{2} & = 0 \end{aligned}$

Circle 2 Question 2

2. If a tangent to the circle $x^{2} + y^{2} = 1$ intersects the coordinate axes at distinct points $P$ and $Q$ , then the locus of the mid-point of $P Q$ is

Answer:

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Circle 2 Question 2

2. If a tangent to the circle x2+y2=1 intersects the coordinate axes at distinct points P and Q, then the locus of the mid-point of PQ 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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2. If a tangent to the circle $x^{2} + y^{2} = 1$ intersects the coordinate axes at distinct points $P$ and $Q$ , then the locus of the mid-point of $P Q$ is