(a) If $$ \frac{a}{4}=\frac{b}{3} $$, find the value of: $$ \frac{a b+a^{2}}{a b-b^{2}} $$ (b) If $$ b $$ is the middle proportion between a and $$ c $$, prove that : $$ \sqrt{\frac{a^{2}+b^{2}}{b^{2}+c^{2}}}=\frac{a}{b} $$ where $$ a, b, c $$ are positive proportion

Question

UpStudy AI · Accepted Answer

### الحل:

#### الجزء (أ):

معطى المعادلة:
$$
\frac{a}{4} = \frac{b}{3}
$$
لنجد قيمة التعبير:
$$
\frac{ab + a^{2}}{ab - b^{2}}
$$

**الخطوات:**

1. **إيجاد علاقة بين $$ a $$ و$$ b $$:**
$$
\frac{a}{4} = \frac{b}{3} \Rightarrow 3a = 4b \Rightarrow a = \frac{4}{3}b
$$

2. **حساب البسط $$ ab + a^{2} $$:**
$$
ab + a^{2} = a b + a^{2} = \left( \frac{4}{3}b \right) b + \left( \frac{4}{3}b \right)^{2} = \frac{4b^{2}}{3} + \frac{16b^{2}}{9} = \frac{12b^{2} + 16b^{2}}{9} = \frac{28b^{2}}{9}
$$

3. **حساب المقام $$ ab - b^{2} $$:**
$$
ab - b^{2} = \left( \frac{4}{3}b \right) b - b^{2} = \frac{4b^{2}}{3} - b^{2} = \frac{4b^{2} - 3b^{2}}{3} = \frac{b^{2}}{3}
$$

4. **حساب النسبة الكلية:**
$$
\frac{ab + a^{2}}{ab - b^{2}} = \frac{\frac{28b^{2}}{9}}{\frac{b^{2}}{3}} = \frac{28}{9} \times \frac{3}{1} = \frac{28}{3}
$$

**الإجابة:**
$$
\frac{ab + a^{2}}{ab - b^{2}} = \frac{28}{3}
$$

---

#### الجزء (ب):

معطى أن $$ b $$ هو النسبة الوسطى بين $$ a $$ و$$ c $$، أي:
$$
a : b = b : c \Rightarrow b^{2} = a c
$$
ونريد إثبات المعادلة:
$$
\sqrt{\frac{a^{2} + b^{2}}{b^{2} + c^{2}}} = \frac{a}{b}
$$

**الخطوات:**

1. **استبدال $$ c $$ بدلالة $$ a $$ و$$ b $$:**
$$
b^{2} = a c \Rightarrow c = \frac{b^{2}}{a}
$$

2. **حساب المقام $$ b^{2} + c^{2} $$:**
$$
b^{2} + c^{2} = b^{2} + \left( \frac{b^{2}}{a} \right)^{2} = b^{2} + \frac{b^{4}}{a^{2}} = b^{2} \left(1 + \frac{b^{2}}{a^{2}}\right) = b^{2} \left( \frac{a^{2} + b^{2}}{a^{2}} \right)
$$

3. **حساب نسبة الكسر تحت الجذر:**
$$
\frac{a^{2} + b^{2}}{b^{2} + c^{2}} = \frac{a^{2} + b^{2}}{b^{2} \left( \frac{a^{2} + b^{2}}{a^{2}} \right)} = \frac{a^{2} + b^{2}}{1} \times \frac{a^{2}}{b^{2}(a^{2} + b^{2})} = \frac{a^{2}}{b^{2}}
$$

4. **أخذ الجذر التربيعي لكلا الطرفين:**
$$
\sqrt{\frac{a^{2}}{b^{2}}} = \frac{a}{b}
$$

**الإجابة:**
$$
\sqrt{\frac{a^{2} + b^{2}}{b^{2} + c^{2}}} = \frac{a}{b}
$$
### الحل: #### الجزء (أ): إذا كان $$\frac{a}{4} = \frac{b}{3}$$، فنجد أن: $$ \frac{ab + a^{2}}{ab - b^{2}} = \frac{28}{3} $$ #### الجزء (ب): إذا كان $$ b $$ هو النسبة الوسطى بين $$ a $$ و$$ c $$، فنثبت أن: $$ \sqrt{\frac{a^{2} + b^{2}}{b^{2} + c^{2}}} = \frac{a}{b} $$

(a) If , find the value of:
(b) If is the middle proportion between a and ,
prove that :
where are positive proportion

Upstudy AI Solution

Answer

الحل:

الجزء (أ):

الجزء (ب):

Solution

الحل:

الجزء (أ):

الجزء (ب):

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(a) If , find the value of: (b) If is the middle proportion between a and , prove that : where are positive proportion

Upstudy AI Solution

Answer

الحل:

الجزء (أ):

الجزء (ب):

Solution

الحل:

الجزء (أ):

الجزء (ب):

Extra Insights

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Latest Algebra Questions

(a) If , find the value of:
(b) If is the middle proportion between a and ,
prove that :
where are positive proportion