Coordinate Geometry

The general equation of a circle with center and radius is: For this circle, the center is , so the equation of the circle becomes:

We are given the equation of one of the tangents from the origin, which is: This equation represents a straight line with slope (since it can be written as ).

Step 1: Find the slope of the radius at the point of tangency
The radius of the circle is always perpendicular to the tangent at the point of contact. Therefore, the product of the slopes of the radius and the tangent is .
If the slope of the tangent is , the slope of the radius is:

Step 2: Find the equation of the radius
The radius passes through the center of the circle, which is , and has a slope of . The equation of the radius is: Simplifying:

Step 3: Find the equation of the second tangent
The second tangent will have a slope that is symmetric to the first tangent with respect to the radius. Since the first tangent has slope , the second tangent will have slope . The equation of the second tangent passing through the origin with slope is: Rearranging:

Answer:

We are given the line , and we need to determine which conic section it touches.

Step 1: The Parabola
For the parabola , the equation can be rewritten as: The line will touch this parabola if it intersects it at exactly one point. Substituting into the equation : Expanding: Simplifying: The discriminant of this quadratic equation is: Since the discriminant is 0, the line touches the parabola at exactly one point.

Step 2: The Ellipse
Rewriting the equation of the ellipse: Substituting into this equation: This gives a quadratic equation that will have two solutions, meaning the line intersects the ellipse at two points. Therefore, the line does not touch the ellipse.

Step 3: The Hyperbola
Rewriting the equation of the hyperbola: Substituting : This gives a quadratic equation that will have two solutions, meaning the line intersects the hyperbola at two points. Therefore, the line does not touch the hyperbola.

Step 4: The Circle
Rewriting the equation of the circle: Substituting : This gives a quadratic equation that will have two solutions, meaning the line intersects the circle at two points. Therefore, the line does not touch the circle.

Answer:

Let the equation of the line be , where and are the points of intersection of the line with the coordinate axes. The radius of the circle circumscribed about the triangle is given by the formula: where is the area of the triangle and is the product of the lengths of the sides of the triangle. However, we can simplify the situation using the concept of tangents.

Step 1: Tangent formula and relationship between the points
We are given that and are the distances of the tangents from the origin to the points and , respectively. The formula for the length of the tangent from the origin to a point is given by: Thus, for points and , the tangent lengths are:

Step 2: Diameter of the circumcircle
The formula for the diameter of the circumcircle is related to the sum of the distances from the origin to the points of tangency. The diameter is simply the sum of the tangent lengths:

Answer:

Given:
Points , , with , , and point on the x-axis.

We want to maximize angle .

Step 1: Slopes of lines AP and BP

Step 2: Use angle between lines formula

Step 3: Maximize the expression
Let:
Using the quotient rule:
Set to find maximum:
Conclusion:
The angle is maximum when:
Correct answer: Option (A)