Ufumbuzi - Mali za ellipses

$$h$$ inawakilisha offset ya x kutoka kwa asili.
$$k$$ inawakilisha offset ya y kutoka kwa asili.
Kupata maadili ya $$h$$ na $$k$$, tumia fomu standard ya duara ya wima:
$$\frac{{\left(x-h\right)}^2}{b^2}+\frac{{\left(y-k\right)}^2}{a^2}=1$$

$$\frac{x^2}{\frac{1}{42}}+\frac{y^2}{2}=1$$
$$h=0$$
$$k=0$$
Center: $$\left(0,0\right)$$

$$a$$ inawakilisha radius ndefu zaidi ya duara, ambayo ni sawa na nusu ya mhimili mkuu.
Hii inaitwa mhimili semi-major.
Kupata thamani ya $$a$$, tumia fomu standard ya duara ya wima:
$$\frac{{\left(x-h\right)}^2}{b^2}+\frac{{\left(y-k\right)}^2}{a^2}=1$$

$$\frac{x^2}{\frac{1}{42}}+\frac{y^2}{2}=1$$
$$a^2=2$$
Chukua mizizi mraba ya pande zote mbili za equation:
$$a=1.414$$

Kwa sababu $$a$$ inawakilisha umbali, ina thamani positive tu.

Katika duara la wima, mhimili mkuu unatembea sawia na mhimili wa y-axis na upita kwenye vertices za duara. Pata vertices kwa kuongeza na kutoa $$a$$ kutoka kwa y-coordinate ($$k$$) ya kituo.

Kupata vertex_1, ongeza $$a$$ kwa y-kiashiria ($$k$$) ya kituo:
Vertex_1: $$\left(h,k+a\right)$$
Kituo: $$\left(h,k\right)=\left(0,0\right)$$
$$h=0$$
$$k=0$$
$$a=1.414$$
Vertex_1: $$\left(0,0+1.414\right)$$
Vertex_1: $$\left(0,1.414\right)$$

Kupata vertex_2, toa $$a$$ kutoka y-kiashiria ($$k$$) ya kituo:
Vertex_2: $$\left(h,k-a\right)$$
Kituo: $$\left(h,k\right)=\left(0,0\right)$$
$$h=0$$
$$k=0$$
$$a=1.414$$
Vertex_2: $$\left(0,0-1.414\right)$$
Vertex_2: $$\left(0,-1.414\right)$$

$$b$$ inawakilisha radius fupi ya ellipse, ambayo ni sawa na nusu ya mhimili mdogo. Hii inaitwa mhimili mdogo.
Kupata thamani ya $$b$$, tumia fomu ya kawaida ya ellipse wima:
$$\frac{{\left(x-h\right)}^2}{b^2}+\frac{{\left(y-k\right)}^2}{a^2}=1$$

$$\frac{x^2}{\frac{1}{42}}+\frac{y^2}{2}=1$$
$$b^2=\frac{1}{42}$$
Chukua karo ya pande zote mbili za equation:
$$b=0.154$$
Kwa sababu b inawakilisha umbali, ina thamani chanya tu.

Katika ellipse wima, mhimili mdogo unakimbia sambamba na x-aksi na kupitia co-vertices ya ellipse.
Pata co-vertices kwa kuongeza na kutoa $$b$$ kutoka x-kiashiria ($$h$$) ya kituo.

Kupata co-vertex_1, ongeza $$b$$ kwa x-kiashiria ($$h$$) ya kituo:
Co-vertex_1: $$\left(h+b,k\right)$$
Kituo: $$\left(h,k\right)=\left(0,0\right)$$
$$h=0$$
$$k=0$$
$$b=0.154$$
Co-vertex_1: $$\left(0+0.154,0\right)$$
Co-vertex_1: $$\left(0.154,0\right)$$

Kiupata co-vertex_2, toa $$b$$ from x-kiashiria ($$h$$) ya kituo:
Co-vertex_2: $$\left(h-b,k\right)$$
Kituo: $$\left(h,k\right)=\left(0,0\right)$$
$$h=0$$
$$k=0$$
$$b=0.154$$
Co-vertex_2: $$\left(0-0.154,0\right)$$
Co-vertex_2: $$\left(-0.154,0\right)$$

Urefu wa mbele ni umbali kutoka kituo cha ellipse kwa kila hatua ya maana na kawaida inawakilishwa na $$f$$.

Kupata $$f$$, tumia formula:
$$f=\sqrt{a^2-b^2}$$
$$a^2=2$$
$$b^2=\frac{1}{42}$$
Chomeka $$a^2$$ na $$b^2$$ katika formula na rahisisha:

$$f=\sqrt{2-\frac{1}{42}}$$

$$f=\sqrt{\frac{83}{42}}$$

$$f=1.406$$

Kwa sababu $$f$$ inawakilisha umbali, ina thamani ya chanya tu.

Katika ellipse ya wima, mhimili kuu unakimbia sambamba na mhimili wa y na kupitia focii.
Pata focii kwa kujumlisha na kutoa $$f$$ kutoka kwa koordinati ya y $$\left(k\right)$$ ya kitovu.

Kupata focus_1, ongeza $$f$$ kwenye koordinati ya y $$\left(k\right)$$ ya kitovu:
Focus_1: $$\left(h,k+f\right)$$
Kitovu: $$\left(h,k\right)=\left(0,0\right)$$
$$h=0$$
$$k=0$$
$$f=1.406$$
Focus_1: $$\left(0,0+1.406\right)$$
Focus_1: $$\left(0,1.406\right)$$

Kupata focus_2, toa $$f$$ kutoka kwa koordinati ya y $$\left(k\right)$$ ya kitovu:
Focus_2: $$\left(h,k-f\right)$$
Kitovu: $$\left(h,k\right)=\left(0,0\right)$$
$$h=0$$
$$k=0$$
$$f=1.406$$
Focus_2: $$\left(0,0-1.406\right)$$
Focus_2: $$\left(0,-1.406\right)$$

Tumia formula ya eneo la duara refu kupata eneo la duara refu:
$$\pi ·a·b$$
$$a=1.414$$
$$b=0.154$$
Weka $$a$$ na $$b$$ kwenye formula na rahisisha:

$$\pi ·1.414·0.154$$

$$\pi ·0.218$$

Eneo ni sawa na $$0.218\pi$$

Ili kupata kuingilio la x-, weka $$0$$ kwa $$y$$ kwenye equation standard ya elipsi na utatue equation ya pili ya daraja kwa $$x$$.
Bonyeza hapa kwa maelezo ya hatua kwa hatua ya equation ya pili ya daraja.

$$\frac{x^2}{\frac{1}{42}}+\frac{y^2}{2}=1$$

$$\frac{x^2}{\frac{1}{42}}+\frac{0^2}{2}=1$$

$$x_1=0.154$$

$$x_2=-0.154$$

Ili kupata kuingilio la y-, weka $$0$$ kwa $$x$$ kwenye equation standard ya elipsi na utatue equation ya pili ya daraja kwa $$y$$.
Bonyeza hapa kwa maelezo ya hatua kwa hatua ya equation ya pili ya daraja.

$$\frac{x^2}{\frac{1}{42}}+\frac{y^2}{2}=1$$

$$\frac{0^2}{\frac{1}{42}}+\frac{y^2}{2}=1$$

$$y_1=1.414$$

$$y_2=-1.414$$

Ili kupata eccentricity tumia formula:
$$\frac{\sqrt{a^2-b^2}}{a}$$
$$a^2=2$$
$$b^2=\frac{1}{42}$$
$$a=1.414$$
Wekeza $$a^2$$ , $$b^2$$ na $$a$$ kwenye formula:

$$\frac{\sqrt{2-\frac{1}{42}}}{1.414}$$

$$\frac{\sqrt{\frac{83}{42}}}{1.414}$$

$$\frac{1.406}{1.414}$$

$$0.994$$

Ubunifu unalingana na $$0.994$$