Question 15 of 70beginner🔧 ApplyVery Short1 mark

Does the point (-2.5, 3.5) lie inside, outside or on the circle x² + y² = 25?

Correct Answer

The given equation of the circle is x2+y2=25x^2 + y^2 = 25.

Comparing this with the standard form (x−h)2+(y−k)2=r2(x-h)^2 + (y-k)^2 = r^2, we get center (0,0)(0,0) and radius r=5r=5.

The distance of the point P(−2.5,3.5)P(-2.5, 3.5) from the center (0,0)(0,0) is calculated as:

d=(−2.5−0)2+(3.5−0)2d = \sqrt{(-2.5 - 0)^2 + (3.5 - 0)^2} d=6.25+12.25=18.5d = \sqrt{6.25 + 12.25} = \sqrt{18.5}

Since 18.5<5\sqrt{18.5} < 5 (because 18.5<2518.5 < 25), the distance of the point from the center is less than the radius.

Hence, the point lies inside the circle.

Exercise: EXERCISE 10.1 | Q: 15 | (Page 6)
For More Understanding

Explanation

To determine the position of a point relative to a circle, we compare the distance of the point from the center with the radius of the circle. The context (Example 3) demonstrates how to find the center and radius from a circle's equation. For the circle x2+y2=25x^2 + y^2 = 25, the center is (0,0)(0,0) and radius is 55. If the distance of the point from the center is less than the radius, the point lies inside the circle.

Solution Steps

  1. Step 1: Identify the center and radius of the circle x2+y2=25x^2 + y^2 = 25. Center = (0,0)(0,0), Radius r=5r = 5.

  2. Step 2: Calculate the distance of the point (−2.5,3.5)(-2.5, 3.5) from the center (0,0)(0,0) using the formula (x2−x1)2+(y2−y1)2\sqrt{(x_2-x_1)^2 + (y_2-y_1)^2}. Distance d=(−2.5)2+(3.5)2=6.25+12.25=18.5d = \sqrt{(-2.5)^2 + (3.5)^2} = \sqrt{6.25 + 12.25} = \sqrt{18.5}.

  3. Step 3: Compare the distance dd with radius rr. Since 18.5<5\sqrt{18.5} < 5, the point lies inside the circle.

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