Question

How do you find an equation for the line tangent to the circle $ {x^2} + {y^2} = 25 $ at the point $ (3, - 4) $ ?

Answer 1

Hint: In this question, we are given the equation of a circle and we have to find the equation of the tangent at the point $ (3, - 4) $ , so to solve this question, we must know what a tangent is and how can we find the equation of a line. A line that is perpendicular to the radius of the circle at a point on the circle is said to be the tangent at that point. Using this relation, we can find the slope of the tangent. As the tangent passes through $ (3, - 4) $ so we can find the equation of the tangent using the passing point and the slope.

Complete step-by-step answer:
The slope of the radius is equal to,
 $ {m_1} = \dfrac{{0 - ( - 4)}}{{0 - 3}} = \dfrac{4}{{ - 3}} $
We know that the product of the slopes of two perpendicular lines is equal to -1, so the slope of the tangent will be negative of the reciprocal of the slope of the radius, that is, the slope of the tangent is –
 $ m = \dfrac{{ - 1}}{{{m_1}}} = \dfrac{{ - 1}}{{\dfrac{4}{{ - 3}}}} = - 1 \times \dfrac{{ - 3}}{4} = \dfrac{3}{4} $
Now the equation of a line passing through point $ ({x_1},{y_1}) $ and having a slope m is given as –
 $ y - {y_1} = m(x - {x_1}) $
So, the equation of tangent is –
 $
  y - ( - 4) = \dfrac{3}{4}(x - 3) \\
   \Rightarrow 4(y + 4) = 3(x - 3) \\
   \Rightarrow 4y + 16 = 3x - 9 \\
   \Rightarrow 3x - 4y - 25 = 0 \;
  $
Hence, the equation for the line tangent to the circle $ {x^2} + {y^2} = 25 $ at the point $ (3, - 4) $ is given as $ 3x - 4y - 25 = 0 $
So, the correct answer is “ $ 3x - 4y - 25 = 0 $ ”.

Note: Comparing the given equation of the circle with the standard equation of the circle, that is, $ {(x - h)^2} + {(y - k)^2} = {r^2} $
We get that the coordinates of the centre of the circle is $ (0,0) $ and its radius is 5 units.
We know two passing points of the radius of the circle, so its slope is given as –
 $ {m_1} = \dfrac{{\Delta y}}{{\Delta x}} = \dfrac{{0 - ( - 4)}}{{0 - 3}} = \dfrac{4}{{ - 3}} $
This is how we find the slope of a line, and we can solve similar questions using this approach.