The angle between a pair of tangents drawn from a point P to parabola y^2=4ax is 45° ,the locus of point P is ?

To find the locus of the point P from which the angle between the tangents drawn to the parabola \(y^2 = 4ax\) is 45°, we can use some properties of conics, specifically parabolas. Let's break down the steps to derive the equation of the locus.

Understanding the Geometry of the Problem

The parabola given is \(y^2 = 4ax\). When we draw tangents from a point P to this parabola, the angle between those tangents can be expressed in terms of the coordinates of point P, say \(P(h, k)\).

Using the Condition for the Angle Between Tangents

The angle θ between the two tangents drawn from a point \((h, k)\) to the parabola can be calculated using the formula:

• If the tangents from point P to the parabola meet at angle θ, then:
• \(\tan(\theta) = \frac{2\sqrt{a^2 + b^2}}{(b - a)}\), where a and b are the slopes of the tangents.

For our case, we know that the angle θ is 45°. Therefore, \(\tan(45°) = 1\). This gives us the equation:

Setting Up the Equation

From the properties of the parabola and the point from which tangents are drawn, we can derive that:

• The equation of tangents from point P to the parabola \(y^2 = 4ax\) can be represented as:
• \(y = mx + \frac{a}{m}\) where m is the slope of the tangent.

For the angle between the tangents to be 45°, we can use the condition:

• The discriminant of the quadratic equation representing the tangents must be zero for them to touch the parabola at one point each:
• Substituting into the condition gives us:
• \(k^2 = 4a(h)\)

Finding the Locus

Now, since the tangents form a 45° angle, we can express the relationship between h and k in terms of the angle condition:

Using the derived equations, we reach the locus of P by equating:

• \(k^2 = 4ah\)
• For the angle to be 45°, we must also have:
• \(k^2 = 4ah\) and the slopes must satisfy the angle condition.

After simplifying, we find that:

The locus of the point P is given by the equation:

Final Locus Equation

Thus, the locus of point P can be expressed as:

\(y^2 = 4ax + 4a^2\)

This equation represents a new parabola that opens to the right, shifted vertically, indicating all possible positions of point P from which tangents to the original parabola can be drawn at a 45° angle.

In summary, by analyzing the geometric properties of tangents and applying the condition for angles, we derive the locus of the point P effectively.