Mensuration

To determine the circumference of the circle passing through the point with two normals given by and , follow these steps: Step 1: Understand the problem
- A normal to a circle is a line that passes through the center of the circle.
- The two given lines and are normals to the circle, so they intersect at the center of the circle.
- The circle passes through the point , which lies on its circumference. Step 2: Find the center of the circle The center of the circle is the point of intersection of the two normals. Solve the system of equations: From equation (2), express in terms of : Substitute into equation (1): Substitute into : Thus, the center of the circle is . Step 3: Find the radius of the circle The radius is the distance between the center and the point on the circumference. Use the distance formula: Step 4: Compute the circumference The circumference of a circle is given by: Substitute : Final Answer: The circumference of the circle is .

To solve this problem, we need to find the area of a right
-angled isosceles triangle formed by a pair of lines through the origin and the line , with the right angle at the origin. Step 1: Understand the problem
The triangle is right
-angled at the origin, and it is isosceles, meaning the two legs of the triangle are equal in length.
The two lines forming the legs of the triangle pass through the origin, and the hypotenuse is the line . Step 2: Find the distance from the origin to the line
The distance from the origin to the line is given by the formula: where is the equation of the line, and is the point. For the line , rewrite it in standard form: Here, , , and . Substituting : Step 3: Relate the distance to the triangle
Since the triangle is right
-angled and isosceles, the two legs of the triangle are equal in length. Let the length of each leg be . The hypotenuse of the triangle is the line , and its length is twice the distance from the origin to the line (because the triangle is isosceles). Thus: For a right
-angled isosceles triangle, the relationship between the legs and the hypotenuse is: Substitute the value of the hypotenuse: Solve for : Step 4: Compute the area of the triangle
The area of a right
-angled isosceles triangle is: Substitute : Final Answer: The area of the triangle is: