answer:Let the four intersection points of mathcal{P} and mathcal{C} be (a, a^2), (b, b^2), (c, c^2), and (d, d^2). We substitute y = x^2 into the equation of the circle: [(x-3)^2 + ((x^2)-5)^2 = 100.] Expanding and simplifying this equation results in a fourth-degree polynomial. Given three roots are -2, 0, and 10, use Vieta's formulas to find the fourth root. Since the sum of the roots equals the coefficient of the x^3 term (which is zero due to symmetry), we get: [ a + b + c + d = 0. ] Given a = -2, b = 0, and c = 10, we solve for d: [ -2 + 0 + 10 + d = 0 Rightarrow d = -8. ] The focus of mathcal{P} is (0, 1/4), so we calculate the distances: - Distance from (0, 1/4) to (-2, 4): sqrt{(-2-0)^2 + (4-1/4)^2} = sqrt{4 + 14.0625} = sqrt{18.0625}. - Distance from (0, 1/4) to (0, 0): sqrt{(0-0)^2 + (0 - 1/4)^2} = 1/4. - Distance from (0, 1/4) to (10, 100): sqrt{(10-0)^2 + (100-1/4)^2} = sqrt{100 + 9801.0625} = sqrt{9901.0625}. - Distance from (0, 1/4) to (-8, 64): sqrt{(-8-0)^2 + (64-1/4)^2} = sqrt{64 + 4050.5625} = sqrt{4114.5625}. Summing these distances: [ sqrt{18.0625} + frac{1}{4} + sqrt{9901.0625} + sqrt{4114.5625} = boxed{text{Sum of the distances}}. ]

answer:Here's a revised version of the paragraph: Samantha, a brilliant scientist, faced numerous challenges, including her deafness. However, she refused to let this obstacle interfere with her work or life. Having mastered the skill of lip reading, she was able to seamlessly interact with her colleagues, who made a conscious effort to include her in all discussions. As a result, her perseverance and determination earned her great respect from her colleagues. I made some changes to improve the flow and cohesion of the paragraph. I combined the first two sentences to immediately convey the challenge Samantha faced, and then showed how she overcame it. I also rephrased some of the sentences to make them more concise and connected, using transitional words like however and as a result to link the ideas together. Additionally, I used more descriptive language, such as mastered the skill and seamlessly interact, to make the paragraph more engaging.

answer:Show, don't tell is a fundamental writing principle that advises writers to convey information through action, dialogue, and sensory details rather than simply stating it outright. It's about creating a more immersive experience for the reader by allowing them to infer and interpret the story, rather than being told what to think or feel. In the context of writing, telling typically involves using straightforward statements or exposition to convey information, such as a character's emotions, traits, or backstory. For example: Samantha was a brilliant scientist. This statement simply tells the reader that Samantha is brilliant, but it doesn't provide any evidence or context to support this claim. On the other hand, showing involves using descriptive language, action, and dialogue to convey the same information in a more subtle and engaging way. For instance: Samantha's eyes lit up as she examined the data, her mind racing with connections and insights that had eluded her colleagues. This revised sentence shows Samantha's intelligence and passion for science through her actions and expressions, allowing the reader to infer her brilliance. By showing rather than telling, writers can create a more engaging, dynamic, and believable story that draws the reader in and encourages them to participate in the narrative. This technique can be applied to various aspects of writing, including character development, plot progression, and emotional resonance.

answer:Transitions are words, phrases, or sentences that connect ideas, sentences, or paragraphs together, helping to create a smooth flow of thought and a cohesive text. They signal the relationships between ideas, such as addition, contrast, time, cause and effect, and more. Effective use of transitions enables readers to follow the writer's argument or narrative with ease. Here are some common transition categories with examples of transitional words and phrases: 1. **Addition**: indicates adding information or ideas - Additionally, furthermore, in addition, moreover, also, besides, equally important, similarly Example: The new policy has been successful in reducing costs. Moreover, it has also improved employee morale. 2. **Contrast**: indicates a shift in perspective or contrast - However, on the other hand, but, in contrast, meanwhile, whereas, although Example: The company's profits have increased significantly. However, the competition has also intensified. 3. **Time**: indicates a change in time or sequence - Meanwhile, next, finally, eventually, previously, simultaneously, soon Example: The team worked on the project for months. Eventually, they submitted their proposal and received funding. 4. **Cause and effect**: indicates a causal relationship between ideas - Therefore, thus, as a result, consequently, accordingly, because, since Example: The company invested heavily in marketing. As a result, their sales increased by 20%. 5. **Comparison**: indicates a comparison or analogy - Similarly, likewise, in the same way, just as, likewise, by the same token Example: The new smartphone is just as powerful as a laptop. Similarly, its price is also comparable to a high-end computer. 6. **Summary**: indicates a summary or conclusion - In conclusion, to summarize, in brief, ultimately, in summary, overall Example: The research findings suggest that climate change is a pressing issue. In conclusion, it is essential to take immediate action to reduce carbon emissions. When using transitions, consider the following tips: - Use them sparingly: Transitions should facilitate the flow of ideas, but overusing them can make the text seem forced or artificial. - Choose the right transition: Select a transition that accurately reflects the relationship between the ideas or sentences. - Vary your transitions: Avoid repetitive use of the same transition; instead, mix and match different types to create a more nuanced and engaging text. - Use transitions to connect paragraphs: Transitions can help link paragraphs together, creating a cohesive and logical structure.