SATHEE: Circle 3 Question 4

Circle 3 Question 4

4. The straight line $x + 2 y = 1$ meets the coordinate axes at $A$ and $B$ . A circle is drawn through $A, B$ and the origin. Then, the sum of perpendicular distances from $A$ and $B$ on the tangent to the circle at the origin is

(2019 Main, 11 Jan I)

(a) $2 \sqrt{5}$

(b) $\frac{\sqrt{5}}{4}$

(d) $\frac{\sqrt{5}}{2}$

Show Answer

Answer:

Correct Answer: 4. (d)

Solution:

According to given information, we have the following figure.

From figure, equation of circle (diameter form) is $(x - 1) (x - 0) + (y - 0) y - \frac{1}{2} = 0$

$\Rightarrow x^{2} + y^{2} - x - \frac{y}{2} = 0$

Equation of tangent at $(0, 0)$ is $x + \frac{y}{2} = 0$

$[∵$ equation of tangent at $(x_{1}, y_{1})$ is given by $T = 0$ .

Here, $T = 0$

$\begin{array}{ll} \Rightarrow x x_{1} + y y_{1} - \frac{1}{2} (x + x_{1}) - \frac{1}{4} (y + y_{1}) = 0] \\ \Rightarrow & 2 x + y = 0 \end{array}$

Now, $A M = \frac{| 2 \cdot 1 + 1 \cdot 0 |}{\sqrt{5}} = \frac{2}{\sqrt{5}}$

$[∵$ distance of a point $P (x_{1}, y_{1})$ from a line

$\begin{aligned} and B N = \frac{∣ 2 \cdot b y + c = 0 is \frac{| a x_{1} + b y_{1} + c |}{\sqrt{a^{2} + b^{2}}}]}{\sqrt{5} \frac{1}{2} ∣} = \frac{1}{2 \sqrt{5}} \\ ∴ A M + B N = \frac{2}{\sqrt{5}} + \frac{1}{2 \sqrt{5}} = \frac{4 + 1}{2 \sqrt{5}} = \frac{\sqrt{5}}{2} \end{aligned}$

Circle 3 Question 4

4. The straight line $x + 2 y = 1$ meets the coordinate axes at $A$ and $B$ . A circle is drawn through $A, B$ and the origin. Then, the sum of perpendicular distances from $A$ and $B$ on the tangent to the circle at the origin is

Answer:

Engineering Preparation

Medical Preparation

NCERT Books & Solutions

Important Resources

Other Resources

Syllabus

Test Platform

Sample Mock Test

Mentorship

NCERT Exercise Video Solution

Circle 3 Question 4

4. The straight line x+2y=1 meets the coordinate axes at A and B. A circle is drawn through A,B and the origin. Then, the sum of perpendicular distances from A and B on the tangent to the circle at the origin is

Answer:

Engineering Preparation

Medical Preparation

NCERT Books & Solutions

Important Resources

Other Resources

Syllabus

Test Platform

Sample Mock Test

Mentorship

NCERT Exercise Video Solution

Menu

4. The straight line $x + 2 y = 1$ meets the coordinate axes at $A$ and $B$ . A circle is drawn through $A, B$ and the origin. Then, the sum of perpendicular distances from $A$ and $B$ on the tangent to the circle at the origin is