Question

Tangents are drawn from the point $\left(17,7\right)$ to the circle $x^2+y^2=169$

Statement I: The tangents are mutually perpendicular.

Statement II: The locus of the points from which mutually perpendicular tangents can be drawn to the given circles is $x^2+y^2=338$

• A. Statement I is correct, Statement II is correct; Statement II is the correct explanation for Statement I
• B. Statement I is correct, Statement II is correct; Statement II is not a correct explanation for Statement I
• C. Statement I is correct, Statement II is correct
• D. Statement I is incorrect, Statement II is correct

Answer 1

The correct option is A

Statement I is correct, Statement II is correct; Statement II is the correct explanation for Statement I

Explanation for the correct option:

Statement I:

Calculating distance $CB$

We know that distance between $2$ points $\left(x_1,y_1\right)$and $\left(x_2,y_2\right)$is $\sqrt{{\left(x_2-x_1\right)}^2+{\left(y_2-y_1\right)}^2}$

$$\begin{gathered} CB=\sqrt{{17}^2+7^2} \\ =\sqrt{338} \end{gathered}$$

Finding the equation of the director circle:

The equation of the director circle of the circle $x^2+y^2=a^2$ is $x^2+y^2=2a^2$
And we also know that the locus of the point of intersection of two perpendicular tangents to a given circle is known as its director circle.

The equation of the director circle in this case is $x^2+y^2=2{\left(13\right)}^2$$=338$The point $\left(17,7\right)$ lies on the director circle since it satisfies the equation of the director circle.Therefore by definition, the tangents are mutually perpendicular

Hence statement I is correct

Statement II:

By definition of the director circle: the locus of the points from which mutually perpendicular tangents can be drawn to the given circles is its director circle

And here, the equation of the director circle is $x^2+y^2=338$

Therefore statement II is correct and it explains statement I

Hence, option (A) is the correct answer.