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The point K(8, -5) lies on the circle with equation $$x^2 + y^2 - 12x - 6y - 23 = 0.$$ Find the equation of the tangent to the circle at K. - Scottish Highers Maths - Question 4 - 2018

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The-point-K(8,--5)-lies-on-the-circle-with-equation--$$x^2-+-y^2---12x---6y---23-=-0.$$---Find-the-equation-of-the-tangent-to-the-circle-at-K.-Scottish Highers Maths-Question 4-2018.png

The point K(8, -5) lies on the circle with equation $$x^2 + y^2 - 12x - 6y - 23 = 0.$$ Find the equation of the tangent to the circle at K.

Worked Solution & Example Answer:The point K(8, -5) lies on the circle with equation $$x^2 + y^2 - 12x - 6y - 23 = 0.$$ Find the equation of the tangent to the circle at K. - Scottish Highers Maths - Question 4 - 2018

Step 1

State centre of circle

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Answer

The given equation of the circle can be rewritten in the standard form. To do this, we rearrange it:

x2−12x+y2−6y=23.x^2 - 12x + y^2 - 6y = 23.

Next, we complete the square for both the x and y terms:

  • For x2−12xx^2 - 12x, we take half of -12, which is -6, and square it to get 36.
  • For y2−6yy^2 - 6y, half of -6 is -3, and squaring it gives us 9.

Thus, we rewrite the equation:

(x−6)2+(y−3)2=68.(x-6)^2 + (y-3)^2 = 68.

The center of the circle is at the point (6, 3).

Step 2

Find gradient of radius

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Answer

The radius connects the center of the circle (6, 3) to the point K(8, -5). The gradient (slope) of the radius can be calculated using the formula:

m=y2−y1x2−x1m = \frac{y_2 - y_1}{x_2 - x_1} where (x1,y1)=(6,3)(x_1, y_1) = (6, 3) and (x2,y2)=(8,−5)(x_2, y_2) = (8, -5).

Calculating,

m=−5−38−6=−82=−4.m = \frac{-5 - 3}{8 - 6} = \frac{-8}{2} = -4.

Step 3

State gradient of tangent

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Answer

The gradient of the tangent line is the negative reciprocal of the radius' gradient. Thus:

mtangent=−1−4=14.m_{tangent} = -\frac{1}{-4} = \frac{1}{4}.

Step 4

State equation of tangent

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Answer

Using the point-slope form of the line equation, which is:

y−y1=m(x−x1),y - y_1 = m(x - x_1), with point K(8, -5) and gradient 14\frac{1}{4}:

y−(−5)=14(x−8).y - (-5) = \frac{1}{4}(x - 8).

Simplifying this:

y+5=14x−2,y + 5 = \frac{1}{4}x - 2, thus:

y=14x−7.y = \frac{1}{4}x - 7.

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