Question

If the tangent at $(1,7)$ to the curve ${x^2} = y - 6$ touches the circle ${x^2} + {y^2} + 16x + 12y + c = 0$ then the value of $c$ is
$(1)85$
$(2)95$
$(3)195$
$(4)185$

Answer 1

Hint: First, we have to define what is slope; the slope is calculated by finding the ratio of the vertical changes in the given x and y-axis and also finding the horizontal changes between any two distinct points on the line.
The vertical changes between the given two points are called the rise, and horizontal changes are known as the run.
Formula used: Tangent of the slope $m = {\left| {\dfrac{{dy}}{{dx}}} \right|_{(x,y)}}$

Complete step-by-step solution:
Let from the given equation the curve equation is ${x^2} = y - 6$.
Now convert the equation into single degree y on the left side, so that we are able to differentiate with the value y to simplify the solution.
Thus, we get ${x^2} = y - 6 \Rightarrow y = {x^2} + 6$.
Differentiate the value y with base x, thus we get $\dfrac{{dy}}{{dx}} = 2x$.
Now we are going to apply the formula of the solution to the diffraction of the function thus we get,
$m = {\left| {\dfrac{{dy}}{{dx}}} \right|_{(1,7)}} \Rightarrow 2x$ (since on the right side of the equation there are only x terms, we are applying the x terms as x is one)
Thus, we get, $m \Rightarrow 2(1) = 2$
Since the equation of the tangent points is $(1,7)$ apply in the given equation we get $y - 7 = 2(x - 1)$
Solving the equation, we get, $y - 7 = 2(x - 1) \Rightarrow 2x - y + 5 = 0$
From the given circle equation as ${x^2} + {y^2} + 16x + 12y + c = 0$ convert this equation according to the plane touches thus we get, ${(x + 8)^2} + {(y + 6)^2} + c - 64 - 36 = 0$
If we simplify the terms and cancel the common values, we get the given,
Thus, we get, ${(x + 8)^2} + {(y + 6)^2} = 100 - c$
Since from this equation, we get the distance as $( - 8, - 6)$ from the equation $2x - y + 5 = 0$
Hence the distance is $d = \left| {\dfrac{{2( - 8) - ( - 6)}}{{\sqrt {4 + 1} }}} \right|$ after simplifying the terms we get, $d = \left| {\sqrt 5 } \right|$
Now equivalent this to the radius we get, Radius $\sqrt {100 - c} = \sqrt 5 $ where c is the constant
Thus, we get $c = 95$ (is the value of the c is ninety-five then both equations are equal)
Hence the value of c is $95$. And this option$(2)95$ is correct.
There is no possibility of getting other options because if the value of c is not $95$ then the radius value will be not equivalent $\sqrt {100 - c} = \sqrt 5 $.

Note: Tangent is the line that the plane curves at the given points in the straight line which gets touched into the curve plane.
Where c is the constant the value of the c does not affect the square root.
$m = {\left| {\dfrac{{dy}}{{dx}}} \right|_{(x,y)}}$ is the general formula for slope where x and y are the axes.