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 $x^{2} + y^{2} = 338$. (1) Statement-1 is True, statement-2 is True; Statement-2 is a correct explanation for Statement-1. (2) Statement-1 is True, Statement-2 is True; Statement-2 is NOT a correct explanation for Statement-1 (3) Statement-1 is True, Statement-2 is False (4) Statement-1 is False, Statement-2 is True.

$x^2 + y^2 = 169 \quad\Rightarrow\quad \text{centre } O(0,0),\ r = 13$

$P(17, 7), \qquad OP^2 = 17^2 + 7^2 = 338$

$\text{S-1: the two tangents from } P \text{ are mutually perpendicular?}$

$\text{S-2: locus of such points is } x^2 + y^2 = 338?$

$OP^2 = 338 > r^2 = 169 \ \Rightarrow\ P \text{ outside}$

$1.\ \text{tangent length and contact points}$

$2.\ \text{angle between the two tangents}$

$3.\ \text{the locus (director circle)}$

$PT = \sqrt{OP^2 - r^2} = \sqrt{338 - 169} = \sqrt{169} = 13$

$\text{tangent at } (x_1, y_1):\ x x_1 + y y_1 = 169$

$P(17,7) \text{ lies on it} \ \Rightarrow\ 17x_1 + 7y_1 = 169$

$\Rightarrow\ T_1(5, 12), \quad T_2(12, -5)$

OT ⊥ PT ⇒ ∠ OTP = 90^∘

$\sin\alpha = \dfrac{OT}{OP} = \dfrac{13}{\sqrt{338}} = \dfrac{1}{\sqrt2}$

$\Rightarrow\ \alpha = 45^\circ$

$\text{angle between tangents} = 2\alpha = 90^\circ$

$\Rightarrow\ \text{Statement-1 is TRUE}$

$\text{director circle}: \ OP^2 = 2r^2$

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

$\Rightarrow\ \text{Statement-2 is TRUE}$

$P(17,7):\ 17^2 + 7^2 = 338 = 2r^2$

$\Rightarrow\ \text{Statement-2 explains Statement-1}$

$\text{Both statements are TRUE; S-2 explains S-1}$

$\boxed{\text{Answer: Option (1)}}$