Point P (2, -7) is the centre of a circle with radius 13 units, PT is perpendicular to chord AB and T = (-2, -4); Calculate the length of :

(i) AT

(ii) AB.

(i) Given:

Radius = PA = PB = 13 units

Distance between the given points = $\sqrt{(x$2 - x1)^2 + (y2 - y1)^2}

Distance between P(2, -7) and T(-2, -4):

$= \sqrt{(-2 - 2)^2 + (-4 + 7)^2}\\[1em] = \sqrt{(-4)^2 + (3)^2}\\[1em] = \sqrt{16 + 9}\\[1em] = \sqrt{25}\\[1em] = 5$

Using Pythagoras theorem in triangle PAT,

PA² = PT² + AT²

⇒ AT² = PA² - PT²

⇒ AT² = 13² - 5²

⇒ AT² = 169 - 25

⇒ AT² = 144

⇒ AT = $\sqrt{144}$

⇒ AT = 12 units

Hence, the value of AT = 12 units.

(ii) We know that the perpendicular from the center of a circle to a chord bisects the chord.

AB = 2AT

= 2 x 12 units

= 24 units

Hence, the length of AB = 24 units.