First slide

Definition of a circle

Question

The equations of the tangents drawn from (7, 1)
to the circle $x^{2} + y^{2} = 25$ are

Moderate

A

$3 x + 4 y - 25 = 0, 4 x - 3 y - 25 = 0$
B

$4 x + 3 y - 31 = 0, 3 x - 4 y - 17 = 0$
C

$3 x - 2 y - 19 = 0, 2 x + 3 y - 17 = 0$
D

none of these

Solution

The equation of any line through (7, 1) is
$y - 1 = m (x - 7) \Rightarrow m x - y - 7 m + 1 = 0$ (i)
the coordinates of the centre and radius of the given circle are
(0, 0) and 5 respectively.
The line (i) will touch the given circle, if,
Length of the perpendicular from the centre = Radius
$\begin{array}{l} \Rightarrow |\frac{m \times 0 - 0 - 7 m + 1}{\sqrt{m^{2} + (- 1)^{2}}}| = 5 \\ \Rightarrow |\frac{1 - 7 m}{\sqrt{m^{2} + 1}}| = 5 \\ \Rightarrow \frac{(1 - 7 m)^{2}}{m^{2} + 1} = 25 \Rightarrow 24 m^{2} - 14 m - 24 = 0 \Rightarrow m = - 3 / 4, 4 / 3 \end{array}$
Substituting the values of m in (i), we obtain
$- \frac{3}{4} x - y + \frac{21}{4} + 1 = 0 and \frac{4}{3} x - y - \frac{28}{3} + 1 = 0$
$\Rightarrow 3 x + 4 y - 25 = 0 and 4 x - 3 y - 25 = 0$
which are the equations of tangents

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