Question 8 of 18intermediate🔧 ApplyNumerical4 marks

If Q(0, 1) is equidistant from P(5, -3) and R(x, 6), find the values of x. Also find the distances QR and PR.

Correct Answer

Step 1: Calculate distance QP Using the distance formula, the distance between points P(5, -3) and Q(0, 1) is: QP = √[(5 - 0)² + (-3 - 1)²] QP = √[5² + (-4)²] QP = √[25 + 16] QP = 41\sqrt{41} units.

Step 2: Calculate distance QR in terms of x Using the distance formula, the distance between points R(x, 6) and Q(0, 1) is: QR = √[(x - 0)² + (6 - 1)²] QR = √[x² + 5²] QR = x2+25\sqrt{x² + 25} units.

Step 3: Find the values of x Since point Q is equidistant from P and R, QP = QR. Therefore, QP² = QR². (41\sqrt{41})² = (x2+25\sqrt{x² + 25})² 41 = x² + 25 x² = 41 - 25 x² = 16 x = ±4. So, the values of x are 4 and -4.

Step 4: Find distance QR As established in Step 1, since Q is equidistant from P, QR = QP. Therefore, QR = 41\sqrt{41} units.

Step 5: Find distance PR We calculate PR for both values of x. Case 1: When x = 4 R is (4, 6). PR = √[(4 - 5)² + (6 - (-3))²] PR = √[(-1)² + (9)²] PR = √[1 + 81] PR = 82\sqrt{82} units.

Case 2: When x = -4 R is (-4, 6). PR = √[(-4 - 5)² + (6 - (-3))²] PR = √[(-9)² + (9)²] PR = √[81 + 81] PR = 162\sqrt{162} = 92\sqrt{2} units.

Exercise: EXERCISE 7.1 | Q: 9 | (Page 7)
For More Understanding

Explanation

This question tests the application of the distance formula in coordinate geometry. The core concept is that 'equidistant' means the distances from a point to two other points are equal. We calculate the distance QP first because all coordinates are known. Then we set up an equation equating QP to QR to solve for the unknown coordinate x. Finally, we calculate the specific distances requested. Note that while QR remains constant for both values of x (due to the square in the formula), PR changes depending on the position of R.

Solution Steps

  1. Step 1: Apply distance formula to find QP using coordinates P(5, -3) and Q(0, 1). Result: 41\sqrt{41}.

  2. Step 2: Apply distance formula to find QR using coordinates R(x, 6) and Q(0, 1). Result: x2+25\sqrt{x² + 25}.

  3. Step 3: Equate QP and QR to solve for x. (41\sqrt{41})² = x² + 25 leads to x = ±4.

  4. Step 4: State QR. Since Q is equidistant, QR = QP = 41\sqrt{41}.

  5. Step 5: Calculate PR for x=4 and x=-4 separately using the distance formula between P and R.

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