**Core Concept**
The patient's symptoms, such as bilateral hip pain and painful limping gait, are suggestive of a condition known as **Legg-Calve-Perthes disease** or **avascular necrosis of the femoral head**. This condition is characterized by the loss of blood supply to the femoral head, leading to its necrosis and subsequent collapse.

**Why the Correct Answer is Right**
The correct answer is **C**. This is because **MRI of the spine** is not a useful investigation in diagnosing Legg-Calve-Perthes disease. The condition primarily affects the femoral head and hip joint, rather than the spine. While MRI is a valuable tool for assessing various musculoskeletal conditions, it is not indicated in this case.

**Why Each Wrong Option is Incorrect**
* **Option A:** **Plain X-ray of the hip** is actually a useful investigation for diagnosing Legg-Calve-Perthes disease, as it can show characteristic changes such as a "crescent sign" or "sclerosis" of the femoral head.
* **Option B:** **Blood tests for rheumatoid factor and ESR** may be useful in evaluating the patient for other conditions that may be contributing to her symptoms, such as rheumatoid arthritis or inflammatory arthritis, but are not directly useful in diagnosing Legg-Calve-Perthes disease.
* **Option D:** **Bone scan** can be useful in detecting areas of abnormal bone metabolism, including those associated with Legg-Calve-Perthes disease, and may be indicated in the diagnostic workup.

**Clinical Pearl / High-Yield Fact**
It's worth noting that Legg-Calve-Perthes disease is more common in children with a family history of the condition and may be associated with other conditions such as **hypothyroidism**, which is present in this patient. Early diagnosis and treatment are crucial in preventing long-term complications.

**Correct Answer:** C. MRI of the spine