Tangents are drawn from the point (17, 7) to the circle x² + y² = 169. STATEMENT-1 : The tangents ar

Q: Wait, why is the Director Circle equation $x^2 + y^2 = 2r^2$ and not just $r^2$?

That is a fantastic question! Think about the geometry. If you are at a point on the circle itself, you can only draw one tangent. As you move further away, the angle between the tangents changes. The Director Circle is the specific boundary where the geometry of the tangents forces them to meet at exactly $90^\circ$. The factor of $2$ comes from the Pythagorean theorem applied to the triangle formed by the center of the circle, the external point, and the point of tangency. It is a beautiful result of the distance from the center being $r\sqrt{2}$.

Q: Can I solve this by finding the slopes of the tangents $m_1$ and $m_2$ and checking if $m_1 \cdot m_2 = -1$?

You absolutely can! That is the 'brute force' method, and it is a great way to verify your understanding. You would write the equation of a line passing through $(17, 7)$ with slope $m$, set the perpendicular distance from the origin to this line equal to the radius $13$, and solve the resulting quadratic equation for $m$. You would find two roots, $m_1$ and $m_2$. If their product is $-1$, the tangents are perpendicular. However, using the Director Circle property is much faster and demonstrates a deeper mastery of coordinate geometry.

Q: Is the Director Circle only for circles centered at the origin?

Great intuition! The formula $x^2 + y^2 = 2r^2$ is specific to circles centered at the origin $(0, 0)$. If the circle were centered at $(h, k)$, the equation of the Director Circle would shift to $(x - h)^2 + (y - k)^2 = 2r^2$. The radius of the Director Circle remains $r\sqrt{2}$ regardless of the center's location, because the perpendicularity condition is an intrinsic property of the circle's size, not its position.

Question

Tangents are drawn from the point $(17, 7)$ to the circle $x^2 + y^2 = 169$. STATEMENT-1 : The tangents are mutually perpendicular. because STATEMENT-2 : The locus of the points from which mutually perpendicular tangents can be drawn to the given circle is $x^2 + y^2 = 338$.

Options:

(A) Statement-1 is True, statement-2 is True; statement-2 is a correct explanation for Statement-1.

(B) Statement-1 is True, Statement-2 is True; Statement-2 is NOT a correct explanation for Statement-1.

(C) Statement-1 is True, Statement-2 is False

(D) Statement-1 is False, Statement-2 is True.

Sigma Percentile · Accepted Answer

### Analyzing the Setup

The given circle is defined by the equation $x^2 + y^2 = 169$. By comparing this to the standard form $x^2 + y^2 = r^2$, we identify the radius $r$ as $\sqrt{169} = 13$.

We are investigating the tangents drawn from an external point $P(17, 7)$ to this circle. Our goal is to determine if these tangents are mutually perpendicular.

### The Director Circle: The Locus of Perpendicularity

The **Director Circle** is the locus of all points from which the two tangents drawn to a given circle are perpendicular to each other. For a circle $x^2 + y^2 = r^2$, the equation of the Director Circle is:

$$x^2 + y^2 = 2r^2$$

This result arises because the center of the circle, the two points of tangency, and the external point form a square when the tangents are perpendicular. The distance from the center to the external point is the diagonal of this square, which is $\sqrt{r^2 + r^2} = r\sqrt{2}$.

### The Verification

Given $r^2 = 169$, the equation of our specific Director Circle is:

$$x^2 + y^2 = 2(169) = 338$$

This confirms that the condition for perpendicular tangents is defined by the equation $x^2 + y^2 = 338$. This matches the criteria provided in Statement-2.

To verify Statement-1, we test if the point $P(17, 7)$ satisfies the equation of the Director Circle. We substitute the coordinates into the left-hand side of the equation:

$$17^2 + 7^2 = 289 + 49 = 338$$

### Conclusion: The Harmony of Logic

Since the calculation yields $338$, which is equal to the right-hand side of the Director Circle equation, the point $P(17, 7)$ lies exactly on the Director Circle.

**Therefore, the tangents drawn from $P(17, 7)$ are mutually perpendicular.**

Statement-1 is true, and Statement-2 provides the correct geometric explanation for this property. This demonstrates the power of identifying hidden geometric structures to solve complex problems efficiently.

Select Answer:

Visualized Solution

The Given Circle

The External Point

The Director Circle Concept

Equation of the Director Circle

Evaluating the Director Circle

Checking Statement-1

Substituting $P (17, 7)$

Calculating the Squares

Final Addition

Drawing the Tangents

Verifying the Angle

Final Conclusion

The Sigma Insight: Director Circle and Family of Circles

Analyzing the Setup

The Director Circle

The Verification

Conclusion

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