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2 years ago

Can someone help me with this question

Mathematics

1 answer:

olga2289 [7]2 years ago

6 0

Answer: $\displaystyle d=3\sqrt{5}$

Step-by-step explanation:

First, we will find the coordinate placement of each point. See attached.

Now, we can use the distance formula.

$\displaystyle d=\sqrt{(x_{2}-x_{1})^2 +(y_{2}-y_{1})^2}$

$d=\sqrt{(6-3)^2 +(-3-3)^2}$

$d=\sqrt{(3)^2 +(-6)^2}$

$d=\sqrt{9+36}$

$d=\sqrt{45}$

$d=\sqrt{9*5}$

$\displaystyle d=3\sqrt{5}$

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Please help! A-level maths.

astra-53 [7]

$x+y=k\implies y=k-x$

Substitute this into the parabolic equation,

$k-x=6-(x-3)^2\implies x^2-7x+(k+3)=0$

We're told the line $x+y=k$ intersects $C$ twice, which means the quadratic above has two distinct real solutions. Its discriminant must then be positive, so we know

$(-7)^2-4(k+3)=49-4(k+3)>0\implies k$

We can tell from the quadratic equation that $C$ has its vertex at the point (3, 6). Also, note that

$-1\le\sin t\le1\implies3\le3+2\sin t\le5\implies3\le x\le5$

and

$-1\le\cos2t\le1\implies2\le4+2\cos2t\le6\implies2\le y\le6$

so the furthest to the right that $C$ extends is the point (5, 2). The line $x+y=k$ passes through this point for $2+5=k\implies k=7$ . For any value of $k$ , the line $x+y=k$ passes through $C$ either only once, or not at all.