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Finding equation of circle (A Maths) 8 mark Question!

Question:

Find the equation of the circle which passes through $A(8,1)$ and $B(7,0)$ and has, for its tangent at $B$ , the line $3x-4y-21=0$ .

Solution:

Recall that the equation of a circle is $(x-a)^2+(y-b)^2=r^2$ , where $(a,b)$ is the centre of the circle, and $r$ is the radius of the circle.

Substituting $A(8,1)$ into the equation, we get:

$\boxed{(8-a)^2+(1-b)^2=r^2}$ — Eqn (1)

Substituting $B(7,0)$ , we get:

$\boxed{(7-a)^2+(0-b)^2=r^2}$ — Eqn (2)

Equating Eqn (1) and Eqn (2), we get

$64-16a+a^2+1-2b+b^2=49-14a+a^2+b^2$ which reduces to

$\boxed{b=8-a}$ — Eqn (3)

after simplification.

Now, we rewrite the equation of the tangent as $\displaystyle y=\frac{3}{4}x-\frac{21}{4}$ (make y the subject)

Hence, the gradient of the normal is $\displaystyle\frac{-1}{\frac{3}{4}}=-\frac{4}{3}$

Let the equation of the normal be $\displaystyle y=-\frac{4}{3}x+c$

Substitute in $B(7,0)$ we get $\displaystyle 0=-\frac{4}{3}(7)+c$

Hence $\displaystyle c=\frac{28}{3}$

Thus equation of normal is $\displaystyle \boxed{y=-\frac{4}{3}x+\frac{28}{3}}$

Since the normal will pass through the centre $(a,b)$ we have

$\boxed{b=-\frac{4}{3}a+\frac{28}{3}}$ — Eqn (4)

Finally, we equate Eqn (3) and Eqn (4),

$\displaystyle 8-a=-\frac{4}{3}a+\frac{28}{3}$

$\displaystyle \frac{1}{3}a=\frac{4}{3}$

$a=4$

$b=8-a=4$

Substituting back into Eqn (1), we get $r=5$

Hence the equation of the circle is:

$\displaystyle\boxed{(x-4)^2+(y-4)^2=5^2}$

Logarithm and Exponential Question: A Maths Question

Question:

Solve $(4x)^{\lg 5} = (5x)^{\lg 7}$

Solution:

$4^{\lg 5}\cdot x^{\lg 5}=5^{\lg 7}\cdot x^{\lg 7}$

$\displaystyle\frac{4^{\lg 5}}{5^{\lg 7}}=\frac{x^{\lg 7}}{x^{\lg 5}}=x^{\lg 7-\lg 5}$

Using calculator, and leaving answers to at least 4 s.f.,

$0.6763=x^{0.1461}$

Lg both sides,

$\lg 0.6763=0.1461\lg x$

$\lg x=\frac{\lg 0.6763}{0.1461}=-1.1626$

$x=10^{-1.1626}=0.0688$ (3 s.f.)

Check answer (to prevent careless mistakes):

$LHS=(4\times 0.0688)^{\lg 5}=0.406$

$RHS=(5\times 0.0688)^{\lg 7}=0.406$

Since LHS=RHS, we have checked that our answer is valid.

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