Answer:
a)
[tex](x - 2) {}^{2} + (x +1) {}^{2} = 20[/tex]
b)
[tex]x - 2y - 14 = 0[/tex]
Step-by-step explanation:
a) the centre is (2,-1)
the equation so far is:
[tex](x - 2) {}^{2} + (x +1) {}^{2} = r {}^{2}[/tex]
because you just change the sign of 2 to -2 and change the sign of -1 to +1
to find the radius:
[tex]r = \sqrt{(change \: in \: x) {}^{2} + (change \: in \: y) {}^{2} }[/tex]
[tex]r = \sqrt{(4 - 2) {}^{2} + ( - 5 - ( - 1)) {}^{2} } [/tex]
[tex]the \: radius \: is \: 2 \sqrt{5} [/tex]
to find diameter square 2√5
[tex](2 \sqrt{5} \:) {}^{2} = 20 \\ the \: equation \: is(x - 2) {}^{2} + (x +1) {}^{2} = 20[/tex]
b) find the gradient between the points (2,-1) and (4,-5)
[tex]gradient = \frac{change \: in \: y}{change \: in \: x} [/tex]
[tex]\: gradient = \frac{ - 5 + 1}{4 - ( - 1)} = - 2[/tex]
the gradient of the tangent is the negative reciprocal of -2. This means you flip it upside down which would be -1/2, then multiply by -1 which is 1/2
the gradient is 1/2
using the gradient and the point (4,-5) to write the equation in the form y=mx+c
here y is -5, m is the gradient (1/2) and x is 4
we need to work out c
[tex]y = mx + c[/tex]
[tex] - 5 = \frac{1}{2} (4) + c[/tex]
[tex]- 5 = 2 + c[/tex]
subtract 2 on both sides
[tex]- 5 - 2 = c[/tex]
[tex]c = - 7[/tex]
now the equation is
[tex]y = \frac{1}{2} x - 7[/tex]
but we need to write it in the form ax+by+c=0
subtract y on both sides:
[tex]\frac{1}{2} x - y - 7 = 0[/tex]
multiply everything by 2 to get rid of the fraction
[tex]x - 2y - 14 = 0[/tex]
that is the final equation
