Question

The length of the tangent from the point (-1,2) to the circle \[{x^2} + {y^2} - 8x + 5y - 7 = 0\]
A. 1
B. 4
C. 2
D. 6

Answer 1

Hint: Bring the equation of the circle in the general form and then find the value of the radius and coordinates of the center from there, then find the distance between the outside point the point on the circle, if you get the distance greater than the radius of the circle then just put the points (-1,2) in the equation of the circle given and square root it, you will easily get the length of the tangent.

Complete step by step answer:
So we are given the equation of the circle as \[{x^2} + {y^2} - 8x + 5y - 7 = 0\]
Which can be written as
\[\begin{array}{l}
\therefore {x^2} + {y^2} - 8x + 5y - 7 = 0\\
 \Rightarrow {x^2} - 8x + 16 + {y^2} + 5y + \dfrac{{25}}{4} = 16 + \dfrac{{25}}{4} + 7\\
 \Rightarrow {\left( {x - 4} \right)^2} + {\left( {y + \dfrac{5}{2}} \right)^2} = \dfrac{{117}}{4}
\end{array}\]
So the center of the circle is \[\left( {4, - \dfrac{5}{2}} \right)\] and the radius is \[\dfrac{{\sqrt {117} }}{2}\] units
We can easily get this by comparing the equation with the general equation of circle which is
\[{\left( {x - h} \right)^2} + {\left( {y - k} \right)^2} = {r^2}\]
Where (h,k) is basically the center of the circle and r is the radius of the circle
Now let us try to find the distance between the center and the other point given by using distance formula which is \[d = \sqrt {{{\left( {x - {x_1}} \right)}^2} + {{\left( {y - {y_1}} \right)}^2}} \]
So we are getting,
\[\begin{array}{l}
 \Rightarrow d = \sqrt {{{\left( { - 1 - 4} \right)}^2} + {{\left( {2 - \left( { - \dfrac{5}{2}} \right)} \right)}^2}} \\
 \Rightarrow d = \sqrt {25 + \dfrac{{81}}{4}} \\
 \Rightarrow d = \dfrac{{\sqrt {181} }}{2}
\end{array}\]
So now we can see that \[\dfrac{{\sqrt {181} }}{2} > \dfrac{{\sqrt {117} }}{2}\] which means that the point is outside the circle
Then the length of the tangent will be
\[\begin{array}{l}
\sqrt {{x^2} + {y^2} - 8x + 5y - 7} \\
 = \sqrt {{{( - 1)}^2} + {2^2} - 8 \times ( - 1) + 5 \times 2 - 7} \\
 = \sqrt {1 + 4 + 8 + 10 - 7} \\
 = \sqrt {16} \\
 = 4
\end{array}\]

So, the correct answer is “Option A”.

Note: Whenever this type of question is given, always check whether the point is inside or outside the circle once you can see that the point is outside the circle then just put the point in the equation of the circle and square root it to get the final value.