SATHEE: Circle 4 Question 17

Circle 4 Question 17

17. Consider the family of circles $x^{2} + y^{2} = r^{2}, 2 < r < 5$ . If in the first quadrant, the common tangent to a circle of this family and the ellipse $4 x^{2} + 25 y^{2} = 100$ meets the coordinate axes at $A$ and $B$ , then find the equation of the locus of the mid-points of $A B$ .

(1999, 5M)

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Answer:

Correct Answer: 17. $4 x^{2} + 25 y^{2} = 4 x^{2} y^{2}$

Solution:

Equation of any tangent to circle $x^{2} + y^{2} = r^{2}$ is

$x \cos θ + y \sin θ = r$

Suppose Eq. (i) is tangent to $4 x^{2} + 25 y^{2} = 100$

or $\frac{x^{2}}{25} + \frac{y^{2}}{4} = 1$ at $(x_{1}, y_{1})$

Then, Eq. (i) and $\frac{x x_{1}}{25} + \frac{y y_{1}}{4} = 1$ are identical

$\begin{aligned} ∴ & \frac{x_{1} / 25}{\cos θ} & = \frac{\frac{y_{1}}{4}}{\sin θ} = \frac{1}{r} \\ \Rightarrow & x_{1} & = \frac{25 \cos θ}{r}, y_{1} = \frac{4 \sin θ}{r} \end{aligned}$

The line (i) meet the coordinates axes in $A (r \sec θ, 0)$ and $β (0, r cosec θ)$ . Let $(h, k)$ be mid-point of $A B$ .

Then,

$h = \frac{r \sec θ}{2} and k = \frac{r cosec θ}{2}$

Therefore, $2 h = \frac{r}{\cos θ}$ and $2 k = \frac{r}{\sin θ}$

$∴ x_{1} = \frac{25}{2 h} and y_{1} = \frac{4}{2 k}$

As $(x_{1}, y_{1})$ lies on the ellipse $\frac{x^{2}}{25} + \frac{y^{2}}{4} = 1$ , we get

$\begin{aligned} \Rightarrow & \frac{1}{25} \frac{625}{4 h^{2}} + \frac{1}{4} \frac{4}{k^{2}} & = 1 \\ or & \frac{25}{4 h^{2}} + \frac{1}{k^{2}} & = 1 \\ 25 k^{2} + 4 h^{2} & = 4 h^{2} k^{2} \end{aligned}$

Therefore, required locus is $4 x^{2} + 25 y^{2} = 4 x^{2} y^{2}$

Circle 4 Question 17

17. Consider the family of circles $x^{2} + y^{2} = r^{2}, 2 < r < 5$ . If in the first quadrant, the common tangent to a circle of this family and the ellipse $4 x^{2} + 25 y^{2} = 100$ meets the coordinate axes at $A$ and $B$ , then find the equation of the locus of the mid-points of $A B$ .

Answer:

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Circle 4 Question 17

17. Consider the family of circles x2+y2=r2,2<r<5. If in the first quadrant, the common tangent to a circle of this family and the ellipse 4x2+25y2=100 meets the coordinate axes at A and B, then find the equation of the locus of the mid-points of AB.

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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17. Consider the family of circles $x^{2} + y^{2} = r^{2}, 2 < r < 5$ . If in the first quadrant, the common tangent to a circle of this family and the ellipse $4 x^{2} + 25 y^{2} = 100$ meets the coordinate axes at $A$ and $B$ , then find the equation of the locus of the mid-points of $A B$ .