Phương trình đường tròn có tâm thuộc đường thẳng \(\Delta: x-2 y=0\), tiếp xúc với đường thẳng \(\Delta^{\prime}: 2 x-y+2=0\) đồng thời đường tròn đi qua điểm \(M(1; 3)\) là:
\((x+2)^{2}+(y+1)^{2}=5\) và \(\left(x+\frac{23}{4}\right)^{2}+\left(y+\frac{23}{8}\right)^{2}=\frac{1445}{64}\).
\((x-2)^{2}+(y-1)^{2}=5\) và \(\left(x-\frac{23}{4}\right)^{2}+\left(y-\frac{23}{8}\right)^{2}=\frac{1445}{64}\).
\((x+1)^{2}+(y-1)^{2}=5\) và \(\left(x-\frac{23}{4}\right)^{2}+\left(y-\frac{23}{8}\right)^{2}=\frac{1445}{64}\).
\((x-2)^{2}+(y-1)^{2}=5\) và \(\left(x-\frac{23}{4}\right)^{2}+\left(y-\frac{23}{8}\right)^{2}=\frac{1885}{16}\).
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Gọi tâm của đường tròn cần tìm là \(I(2 t; t) \in \Delta: x-2 y=0\)
Theo giả thiết, ta có:
\(M I=d\left(I; \Delta^{\prime}\right) \Leftrightarrow \sqrt{(2 t-1)^{2}+(t-3)^{2}}=\frac{|2.2 t-t+2|}{\sqrt{5}}\)
\(\Leftrightarrow \sqrt{5 t^{2}-10 t+10}=\frac{|3 t+2|}{\sqrt{5}} \Leftrightarrow 8 t^{2}-31 t+23=0 \Leftrightarrow\left[\begin{array}{c}t=1 \\ t=\frac{23}{8}\end{array}\right.\)
Với \(t=1\) thì đường tròn cần tìm có tâm \(I(2; 1)\), bán kính \(R=I M=\sqrt{5}\), và có phương trình là: \((x-2)^{2}+(y-1)^{2}=5\)
Với \(t=\frac{23}{8}\) thì đường tròn cần tìm có tâm \(I\left(\frac{23}{4}; \frac{23}{8}\right)\), bán kính \(R=I M=\frac{17 \sqrt{5}}{8}\), và có phương trình là: \(\left(x-\frac{23}{4}\right)^{2}+\left(y-\frac{23}{8}\right)^{2}=\frac{1445}{64}\)
Vậy có hai đường tròn thỏa mãn yêu cầu bài toán là:
\((x-2)^{2}+(y-1)^{2}=5\) và \(\left(x-\frac{23}{4}\right)^{2}+\left(y-\frac{23}{8}\right)^{2}=\frac{1445}{64}\).
Đề Thi Tham Khảo Đánh Giá Năng Lực Năm 2024 – ĐHQG Hà Nội – Đề Số 01 là bài kiểm tra toàn diện và khoa học, giúp học sinh thể hiện năng lực giải quyết vấn đề, tư duy sáng tạo và phân tích logic. Với ba phần thi chính: Toán Học Và Xử Lí Số Liệu, Văn Học - Ngôn Ngữ, và Khoa Học/Tiếng Anh, đề thi không chỉ dừng lại ở việc kiểm tra kiến thức cơ bản mà còn yêu cầu thí sinh phát triển khả năng lập luận và ứng dụng thực tiễn. Đặc biệt, phần thi Khoa Học cho phép lựa chọn giữa Vật Lí, Hóa Học, Sinh Học, Lịch Sử, Địa Lí tạo điều kiện cho thí sinh phát huy thế mạnh cá nhân.
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